JERUSALEM — Israel's Cabinet overwhelmingly backed a deal Sunday with Hezbollah to swap a Lebanese prisoner convicted of a brutal 1979 attack for the bodies of two captured soldiers who were declared dead earlier in the day.

The proposed deal also needs the approval of the Lebanese militant group's secretive, decision-making Shura Council.

The agreement had sparked a fierce public debate over whether Israel would be giving up too much or carrying out its highest commitment to its soldiers to do everything possible to bring them home if they fell into enemy hands.

The Cabinet debated the deal for nearly six hours before voting 22-3 to back the agreement, said Olmert's spokesman, Jacob Galanti. There was no immediate reaction from the soldiers' families or Hezbollah.

The deal would have Hezbollah return two soldiers it captured in a July 2006 cross-border raid that sparked a 34-day war. Israel would release Samir Kantar, imprisoned for a 1979 attack etched in the Israeli psyche as one of the cruelest in the nation's history.

Hezbollah had offered no sign that Ehud Goldwasser and Eldad Regev were alive and the Red Cross was never allowed to see them.

Ahead of the vote, Prime Minister Ehud Olmert said for the first time that Israel has concluded the two soldiers were dead — killed during the raid or shortly after.

"We know what happened to them," Olmert said, according to a prepared statement given to the Cabinet and released by his office. "As far as we know, the soldiers Regev and Goldwasser are not alive."

Goldwasser's wife, Karnit, who has traveled the world to press leaders to push for her husband's release, said she was trying to come to terms with the prime minister's declaration.

"My heart aches. It is very difficult for me. I am very tired, drained inside," she told reporters. "All I want to do is to digest things, try to understand what happened ... to rest a bit ... to have my pain."

In Beirut, Lebanon, Hezbollah said the Israeli decision reflected the guerrilla group's strength.

"What happened in the prisoners issue is a proof that the word of the resistance is the most faithful, strongest and supreme," the group's Al-Manar TV quoted Hezbollah's Executive Council chief Hashem Safieddine as saying.

Israeli media said the soldiers' bodies would be sent to Germany and identified by Israel before Kantar is released. The identification process and swap are expected to take 10 to 14 days, Israeli media reported.

Germany has been trying to mediate a prisoner exchange since Israel's war with Lebanon ended in August 2006.

Kantar is serving multiple life terms in the infiltration attack on a northern Israeli town. Witnesses said Kantar — then 16 — shot Danny Haran in front of his 4-year-old daughter, then smashed her skull against a rock with his rifle butt, killing her, too.

During the attack, Haran's wife accidentally smothered their 2-year-old daughter in a frantic attempt to keep her quiet so Kantar and his comrades wouldn't find them in their apartment's crawl space. Two Israeli policemen also were killed. Kantar denies killing the 4-year-old.

As part of the latest agreement, Israel will also receive a report on missing Israeli airman Ron Arad, whose plane crashed in Lebanon in 1986, and body parts of other Israeli soldiers.

In addition to Kantar, Lebanon will receive four imprisoned Hezbollah fighters, a dozen bodies, most of them Hezbollah militants, and an unspecified number of Palestinian prisoners. Hezbollah had demanded the release of hundreds of Palestinian prisoners.

Critics have argued that swapping bodies for Kantar would offer militant groups an even greater incentive to capture soldiers and less of a reason to keep captives alive.

The debate over the deal taps into a military ethos that runs deep within Israeli society, where most young men and many young women perform compulsory service. Soldiers go out to battle with the understanding they won't be left behind in the field.

The controversy also has weighed the immediacy of the Regev and Goldwasser families' anguish against the pain suffered the family of Kantar's victims. The woman whose family was killed by Kantar, Smadar Haran Kaiser, has in the past opposed his release.

On Sunday, she said she was devastated by the Cabinet's decision, but understood it.

"The despicable murderer Kantar was never my own personal prisoner, but the state's prisoner," she told a news conference. "Even if my soul should be torn, and it is torn, my heart is whole."

Israeli newspapers splashed pictures of the soldiers, their families and military comrades on their front pages.

A recent poll by Israel's Dahaf Research Institute showed that 65 percent of those questioned said Kantar should be released in exchange for the two soldiers held by Hezbollah, even if it was not known whether they are dead or alive.

The soldiers' families had mounted a concerted public campaign to get the government to approve the deal. Family and friends demonstrated outside Olmert's office while the ministers were deliberating.

Karnit Goldwasser said troops would be less willing to fight for their country if they sensed their country had wavered in its commitment to its soldiers.

Some Cabinet ministers took the same view. "I believe in this deal with all my heart. There's no room for hesitation, not to agree to the deal is to erase our obligation to bring back every soldier," Cabinet Minister Meir Sheetrit said ahead of the meeting.

Other politicians were afraid the emotional appeals of the soldiers' families could lead the government to bend sacred principles.

"If they are dead, I certainly oppose this deal," dovish lawmaker Yossi Beilin told Israel Radio. "The principle must be releasing live prisoners for live hostages, and releasing bodies in return for the fallen."

In addition to the two captured soldiers held in Lebanon, Israel is trying to win back a third soldier captured by Palestinian militants in a June 2006 cross-border raid from the Gaza Strip.

Sgt. Gilad Schalit has sent letters and an audio tape to his parents and is believed to be alive, though he has not been seen since his capture and the Red Cross has not been permitted to visit him.