After more than 15 months of nearly uninterrupted fighting in Gaza, diplomats trying to broker a cease-fire between Israel and Hamas expressed cautious hope on Tuesday that a deal could finally be within reach.

But that optimism was tempered — by the uncertainty of whether both sides would ultimately agree, by what details could still change and by the experience of months past in which cease-fire talks repeatedly collapsed.

“We believe that we are at the final stages, but until we have an announcement — there will be no announcement,” Majed al-Ansari, the foreign ministry spokesman for one of the mediating countries, Qatar, told reporters on Tuesday.

Here’s what we know about the potential cease-fire agreement.

What is the state of the talks?

The U.S. secretary of state, Antony J. Blinken, said on Tuesday that Israel and Hamas were “right on the brink” of agreeing to a deal to pause the fighting in Gaza and release hostages held there in exchange for Palestinians held in Israeli jails.

He and diplomats from other mediating countries, including Qatar and Egypt, have for months failed to reach a breakthrough in talks, but they have made progress quickly in recent weeks.

Mediators “managed to minimize a lot of the disagreements between both parties,” Mr. al-Ansari said, adding that they were focused on “the final details of reaching an agreement.”

“On Sunday, the United States, Qatar and Egypt put forward a final proposal,” Mr. Blinken said. “The ball is now in Hamas’ court.”

Officials in both the Israeli government and Hamas have suggested that they are ready to move forward if the other side signs off.

On Monday, a Hamas official said a deal was possible in the coming days as long as Israel did not suddenly change its positions. On Tuesday, an Israeli official said Israel was ready to close the deal and was waiting for Hamas to make a decision.

Hamas officials negotiating in Qatar must obtain the consent of the group’s remaining military commanders in Gaza for the deal. Those commanders include Mohammad Sinwar, whose brother led the group before being killed by Israel in September. Because of their need to to keep their locations secret, communicating with them can be difficult, causing delays.

It was still not clear whether Mr. Sinwar had conveyed his position toward the cease-fire proposal to Hamas leaders in Qatar.

What does the proposal say?

Given the sensitive, continuing negotiations, officials have been wary of describing the proposed agreement except in broad terms.

The framework of the deal was heavily inspired by previous proposals discussed in May and July, said one diplomat familiar with the talks, speaking on condition of anonymity to discuss the volatile negotiations. Those proposals detailed a three-stage cease-fire in which Israeli troops would gradually withdraw from Gaza, as Hamas released hostages in exchange for Palestinians jailed by Israel.

Israeli officials hope to secure the release of at least some of the approximately 100 hostages who have been held in Gaza since the Hamas-led attack on southern Israel in October 2023 that started the war. About 35 of the remaining hostages are presumed dead by the Israeli authorities.

Hamas leaders want to end Israel’s military campaign, secure the entry of materials for reconstruction and gain the freedom of Palestinian prisoners held in Israel.

During the first phase of the proposed cease-fire — which would last roughly six weeks — Hamas would release 33 named hostages, most of whom Israel believes are alive, said an Israeli official, who requested anonymity to discuss the sensitive talks. Israel is willing to release hundreds of Palestinian prisoners in exchange, the official said, but the number depends on how many of the hostages are still alive.

Why have talks moved fast in recent weeks?

Some officials have suggested that the change in U.S. administrations, set to take place on Jan. 20, put pressure on Israel and Hamas to accelerate their decision-making after months of delay.

The incoming U.S. president, Donald J. Trump, has warned that there will be “all hell to pay” unless the hostages are freed by the time he becomes president. Steve Witkoff, his pick for Middle East envoy, has also made trips to Qatar and Israel.

The talks have also gained momentum since Israel reached a separate cease-fire agreement with the Lebanese militant group Hezbollah, which began firing rockets into Israel immediately after the Hamas-led attack of October 2023. Battered badly by its escalating conflict with Israel, Hezbollah agreed to a cease-fire with Israel in November, a deal that helped isolate Hamas.

Deaths from bombs and other traumatic injuries during the first nine months of the war in Gaza may have been underestimated by more than 40 percent, according to a new analysis published in The Lancet.

The peer-reviewed statistical analysis, led by epidemiologists at the London School of Hygiene and Tropical Medicine, used modeling in an effort to provide an objective third-party estimate of casualties. The United Nations has relied on the figure from the Hamas-led Ministry of Health, which it says has been largely accurate, but which Israel criticizes as inflated.

But the new analysis suggests the Hamas health ministry tally is a significant undercount. The researchers concluded that the death toll from Israel’s aerial bombardment and military ground operation in Gaza between October 2023 and the end of June 2024 was about 64,300, rather than the 37,900 reported by the Palestinian Ministry of Health.

The estimate in the analysis corresponds to 2.9 percent of Gaza’s prewar population having been killed by traumatic injury, or one in 35 inhabitants. The analysis did not account for other war-related casualties such as deaths from malnutrition, water-borne illness or the breakdown of the health system as the conflict progressed.

The study found that 59 percent of the dead were women, children and people over the age of 65. It did not establish what share of the reported dead were combatants.

Mike Spagat, an expert on calculating casualties of war who was not involved in this research, said the new analysis convinced him that Gaza casualties were underestimated.

“This is a good piece of evidence that the real number is higher, probably substantially higher, than the Ministry of Health’s official numbers, higher than I had been thinking over the last few months,” said Dr. Spagat, who is a professor at Royal Holloway College at the University of London.

But the presentation of precise figures, such as a 41 percent underreported mortality, is less useful, he said, since the analysis actually shows the real total could be less than, or substantially more. “Quantitatively, it’s a lot more uncertain than I think comes out in the paper,” Dr. Spagat said.

The researchers said their estimate of 64,260 deaths from traumatic injury has a “confidence interval” between 55,298 and 78,525, which means the actual number of casualties is likely in that range.

If the estimated level of underreporting of deaths through June 2024 is extrapolated out to October 2024, the total Gazan casualty figure in the first year of the war would exceed 70,000.

“There is an importance to war injury deaths, because it speaks to the question of whether the campaign is proportional, whether it is, in fact, the case that sufficient provisions are made to to avoid civilian casualties,” said Francesco Checchi, an epidemiologist with an expertise in conflict and humanitarian crises and a professor at the London School of Hygiene and Tropical Medicine who was an author on the study. “I do think memorializing is important. There is inherent value in just trying to come up with the right number.”

The analysis uses a statistical method called capture-recapture analysis, which has been used to estimate casualties in other conflicts, including civil wars in Colombia and Sudan.

For Gaza, the researchers drew on three lists: The first is a register maintained by the Palestinian Ministry of Health, which mainly comprises the dead in hospital morgues and estimates of the number of unrecovered people buried in rubble. The second is deaths reported by family or community members through an online survey form the ministry established on Jan. 1, 2024, when the prewar death registration system had broken down. It asked Palestinians inside and outside Gaza to provide names, ages, national ID number and location of death for casualties. The third source was obituaries of people who died from injuries that were published on social media, which may not include all of the same biographical details and which the researchers compiled by hand.

The researchers analyzed these sources to look for individuals who appear on multiple lists of those killed. A high level of overlap would have suggested that few deaths were uncounted; the low amount they found suggested the opposite. The researchers used models to calculate the probability of each individual appearing on any of the three lists.

“Models enable us to actually estimate the number of people who have not been listed at all,” Dr. Checchi said. That, combined with the listed number, gave the analysts their total.

Patrick Ball, director of research at the Human Rights Data Analysis Group, and a statistician who has conducted similar estimates of violent deaths in conflicts in other regions, said the study was strong and well reasoned. But he cautioned that the authors may have underestimated the amount of uncertainty caused by the ongoing conflict.

The authors used different variations of mathematical models in their calculations, but Dr. Ball said that rather than presenting a single figure — 64,260 deaths — as the estimate, it may have been more appropriate to present the number of deaths as a range from 47,457 to 88,332 deaths, a span that encompasses all of the estimates produced by modeling the overlap among the three lists.

“It’s really hard to do this kind of thing in the middle of a conflict,” Dr. Ball said. “It takes time, and it takes access. I think you could say the range is larger, and that would be plausible.”

While Gaza had a strong death registration process before the war, it now has only limited function after the destruction of much of the health system. Deaths are uncounted when whole families are killed simultaneously, leaving no one to report, or when an unknown number of people die in the collapse of a large building; Gazans are increasingly buried near their homes without passing through a morgue, Dr. Checchi said.

The authors of the study acknowledged that some of those assumed dead may in fact be missing, most likely taken as prisoners in Israel.

Roni Caryn Rabin and Lauren Leatherby contributed reporting.

President Biden will remove Cuba from a list of state sponsors of terrorism, U.S. officials announced on Tuesday, as a part of a deal that is expected to free protesters jailed during a sweeping 2021 crackdown on dissent by the Communist government.

Removing Cuba from the list would typically raise expectations of improved relations between Washington and Havana and renewed hope for economic investment in Cuba.

But Mr. Biden’s decision is only the latest in a series of conflicting U.S. approaches to Cuba by different administrations. The Obama administration made the decision to remove the terrorism designation for Cuba as part of a broader effort to normalize relations with the nation. But days before President Donald J. Trump left office in 2021, his administration placed Cuba back on the list.

Mr. Biden’s decision honored the “wisdom and counsel that has been provided to him by many world leaders, especially in Latin America, who have encouraged him to take these actions, on how best to advance the human rights of the Cuban people,” according to a statement from the White House press secretary, Karine Jean-Pierre.

Mr. Trump’s pick to be secretary of state in his new term, Senator Marco Rubio of Florida, is likely to oppose Mr. Biden’s 11th-hour actions on Cuba. Mr. Rubio’s family fled the island for the United States before Fidel Castro’s revolution seized power in 1959, and he has long been one of the Republican Party’s leading advocates for a hard-line stance against Cuba.

Mauricio Claver-Carone, who was recently named Mr. Trump’s envoy for Latin America, scoffed at the announcement. “Whether Venezuela last year or now Cuba, the Biden administration seems to love fake deals that favor authoritarian anti-American regimes,” he said, referring to a 2023 prisoner swap with Venezuela.

Asked whether the Trump administration would immediately repeal the decision, he said, “There’s a process, so it will take time, but in the meantime we can take other measures that will have even greater impact.”

The decision by Mr. Biden is the latest in a number of environmental, immigration and foreign policy actions taken in the final weeks of his presidency before Mr. Trump returns to the White House.

If upheld, Mr. Biden’s move could help Cuba’s battered economy. Being placed on a terrorism list opens up a nation to economic penalties, including a restriction on accessing U.S. aid and a ban on defense exports and sales, according to the State Department. Major banks stopped doing business with Cuba because it was too much trouble to make sure that the country satisfied all the requirements to legally do business there. Still, even without the terrorist designation, Cuba would still need to contend with the trade embargo imposed by the United States.

John S. Kavulich, president of the U.S.-Cuba Trade and Economic Council, a business group, said American companies would not make any changes to their stance on Cuba, because they know the fragility of the relationship between the two nations. He said the move would probably be reversed as soon as Mr. Rubio “hands President Trump a pen.”

“Any quid obtained by the Biden-Harris administration is not worth the quo,” he said.

Removing Cuba from the list leaves only North Korea, Iran and Syria on it.

The Biden administration’s announcement included measures that would ease economic penalties against Cuba. U.S. officials said they would waive a provision of a 1996 law that allowed people to file lawsuits in U.S. courts over property expropriated by the Cuban government. The Biden administration will also lift a Trump-era national security memorandum that imposed sanctions on Cuban military entities — which oversee the hotel industry — and made it difficult for U.S. tour operators to bring in large groups.

The U.S. officials, speaking on the condition of anonymity to preview Mr. Biden’s decision, said the United States was taking the action as part of an effort, pushed by the Catholic Church, to free political prisoners.

Hundreds of protesters were detained in July 2021 in a brutal crackdown following one of the largest demonstrations in Cuba since the Communist government took power roughly six decades earlier. The demonstrators were angry over the nation’s lack of freedom and severe economic decline.

Human rights groups say that some of those arrested during and after the protests have been tortured and that many have been sentenced to long prison terms after unfair trials. There are about 1,000 political prisoners serving time in Cuba, and about half of their cases were related to the July 2021 protest, human rights groups said.

In a statement, Cuba’s Foreign Ministry said President Miguel Díaz-Canel Bermúdez had made the decision to “gradually” release 553 people “who had been convicted, in accordance with the due process, for several crimes that are punishable by law” after meeting with Pope Francis. The move was made in recognition of the Catholic Church Jubilee, a special year of mercy and forgiveness.

Alejandro González Raga, a dissident who was released in a 2008 accord and sent to Spain, stressed that at least four of Cuba’s main opposition leaders also remained imprisoned, and that they and all remaining political prisoners should be released without any conditions or being forced into exile.

But some in Cuba expressed doubt over the prospect.

“I’ll believe it when I see it,” said Eloy Pedroso, whose son, also called Eloy, was condemned to five years in prison after he appeared on national television at age 18 throwing stones at the police.

In a social media post, Mr. Díaz-Canel thanked all those who had contributed to the decision to remove Cuba from a list he said it should never have been on, which “has a high cost for Cuban families.”

“We will continue to confront and denounce the economic war and the actions of interference, disinformation and discredit financed with U.S. federal funds,” he said.

Many experts have long criticized Cuba’s inclusion on the list of state sponsors of terrorism, and blamed that status for causing a dire financial situation that spurred a humanitarian and migration crisis on the island. The Cuban government has struggled in recent months to keep electricity running and stores stocked.

“The statute that creates the terrorism list specifies giving material support to terrorists or harboring terrorists who are actively engaged in terrorism while you are harboring them,” said William M. LeoGrande, a Cuba expert at American University. “Cuba just hasn’t done those things.”

Several U.S. fugitives do reside in Cuba, but they should not count, because they are Americans who had been involved in politically motivated violence in the United States, not international terrorists, Mr. LeoGrande said.

Another reason cited for keeping Cuba on the list was its refusal to extradite members of a Colombian rebel group, the National Liberation Army, also known as the E.L.N. The rebels received safe passage to Cuba to participate in peace talks, which later broke down. But the president of Colombia, Gustavo Petro, recently asked the Biden administration to remove Cuba from the list, which was one of the key factors that motivated the Biden administration, according to a U.S. official.

Juan Pappier, deputy director of Human Rights Watch’s Americas division, said Mr. Biden’s move came too late.

“Biden made a mistake of putting Cuba in the back seat and failing to overturn the pernicious policies of the Trump administration,” Mr. Pappier said.

He said Mr. Biden delayed the decision out of fear of alienating voters in Florida ahead of the U.S. presidential election.

Migdalia Gutiérrez’s son, Brusnelvis Cabrera Gutiérrez, was sentenced to 10 years for sedition at age 20 for throwing stones, a charge he denies.

“It’s the best that could happen. It’s brilliant!” she said of the anticipated prisoner release. “I want him liberated. I have hope. How am I not going to have hope?”

Ed Augustin contributed reporting from Havana.