The meeting — the second between Sharon and Abbas in two weeks — was called ahead of next week's summit in Jordan with President Bush on the U.S. "road map" for Middle East peace.
The meeting came after the Islamic militant group Hamas held out the possibility of a cease-fire to end attacks that have killed hundreds of Israelis in 32 months of violence.
In the nearly 3-hour meeting Thursday night, Sharon told Abbas that as a good faith measure he would unilaterally end the 2-week-old closure on the West Bank by Saturday and allow some Palestinian workers to enter Israel. Israel also would ease some restrictions on roadblocks around Palestinian towns and planned to release some prisoners, according to Israeli and Palestinian officials.
Sharon demanded Abbas "act decisively to stop terrorism, ... dismantling terror organizations, arresting terrorists, confiscating illegal weapons, stopping incitement and creating an atmosphere of peace," according to a statement from Sharon's office.
"The prime minister emphasized that only after liquidation of terrorism and a real struggle against it will it be possible to achieve the hoped-for peace," it said.
In exchange for the crackdown, the Israelis would pull their forces out of some areas of the West Bank and Gaza and allow the Palestinians to take responsibility of security there, the statement said.
But CBS News Correspondent Richard Roth reports that just hours before the prime ministers met, a handful of Israelis was claiming sovereignty on a new patch of the Palestinian West Bank.
Sharon's government did nothing to discourage the newest settlers and some Israelis say that's no surprise.
"He cherishes them. He tells them, 'You are the best of the best. You are the creme de la crème.' Now he's going to tell them you have to leave your houses? I doubt it," says Akiva Eldar, a columnist for Israel's Ha'aretz newspaper.
Palestinian Foreign Minister Nabil Shaath said the Palestinians would be ready to take responsibility for security in areas of the West Bank and Gaza Strip "after reaching an agreement on a cease fire with Hamas and the other Palestinian groups. Now we will try our best to reach this agreement ... and I'm optimistic."
Abbas has been trying to persuade violent groups to accept a cease-fire in the struggle against Israel, but Israel insists on a crackdown, the word used in the U.S.-backed "road map" peace plan, and destruction of what Israel calls the "terrorist infrastructure."
Abbas said in comments published Thursday that Hamas could agree as early as next week to halt deadly attacks on Israelis. The leader of the Islamic Jihad militant group also offered a conditional cease-fire.
However, in a statement faxed to AP in Gaza, the Al Aqsa Martyrs' Brigades, linked to the mainstream Fatah movement, said it would continue attacks "in any time and in any place while the occupiers are taking our land, our prisoners are still in prison and our refugees are living outside their homeland."
The first phase of the three-year "road map" for ending the bloodshed and creating a Palestinian state begins with statements from the Israelis and Palestinians renouncing violence and recognizing each other's rights to security and statehood.
The three-phase plan calls for parallel steps by the two sides, but Israel has demanded a crack down before the rest of the plan is implemented.
Abbas told the Yediot Ahronot daily that he was close to reaching a cease-fire agreement with Hamas that would commit the group to "stop terrorism" against Israelis both in Israel and in the West Bank and Gaza.
Abbas met last week with Hamas and has another meeting scheduled for next week.
"I believe that next week I will reach a cease-fire agreement with Hamas," he said. Abbas said he also hoped to reach agreement with a second militant group, Islamic Jihad.
Hamas official Abdel Aziz Rantisi told AP on Thursday the group was considering ending attacks against civilians if Israel halted its military operations in the West Bank and Gaza, but it would continue to "resist the occupation" and would not be disarmed.
"We have no intention of giving up our weapons," he said. "To stand with empty hands facing an enemy that tries to target us all the time, that's impossible."
Islamic Jihad leader Abdullah Shami told AP his group was willing to consider an indefinite cease-fire "if the enemy is committed to not targeting our people ... as well as releasing Palestinian prisoners."
Military operations against Islamic militants continued Thursday. Israeli troops moved into the West Bank city of Jenin and killed an Islamic Jihad member during an exchange of gunfire, Palestinian witnesses and the army said.
In Gaza late Thursday, Israeli forces fired a tank shell at the town of Deir el-Balah, killing a Palestinian who was planting a bomb near a Jewish settlement, residents said. The military said soldiers did not open fire in the area. Earlier, in nearby Khan Younis, Israeli troops shot and killed Hamas activist Jihad Qidra and arrested 29 suspected militants, including seven Hamas members, the military said.