Under enormous pressure from the Egyptians and Americans, Hamas last night gave approval in principle to the Egyptian mediator, deputy intelligence chief Mustafa Albuhaeiry, for an end to terrorist attacks and military activities as part of a general cease-fire.

Albuhaeiry arrived in Gaza yesterday afternoon for a series of meetings with Palestinian factions, starting with the Fatah leadership in the Strip. From them he received an agreement for a cease-fire based on the Palestinian Authority's acceptance of the road map.

He then went to Sheikh Ahmed Yassin's home for a meeting with the entire Hamas leadership - including Abdel Aziz Rantisi, who survived an Israeli assassination attempt last week.

Palestinian sources said that in the last 48 hours Hamas had handed position papers to the Egyptians accepting a cease-fire and a cessation of military activity. Negotiations were still underway for conditions that would include a guarantee of Hamas' future as a political movement, a timetable framework for the cease-fire, and a guarantee that PA security services would not arrest Hamas activists in the future.

Albuhaeiry, who is tipped to become the next chief of intelligence in Egypt and who held the portfolio in the Egyptian intelligence services for handling Islamic extremists in Egypt, also sent invitations to Hamas leaders based in Damascus to attend meetings in Cairo on the cease-fire.

The Albuhaeiry meetings in Gaza are part of an overall Egyptian-American push for a cease fire, with pressure on Hamas, Islamic Jihad, the various democratic and popular fronts and the other Palestinian factions - including the Fatah's Tanzim.

Palestinian sources said Hamas was warned that if it did not accept a cease-fire it would face combined Israeli-Palestinian pressure aimed at eliminating the organization.

At the meeting in Yassin's house, the spiritual leader of the Hamas said his movement accepts the security principles that were worked out by the PA for accepting security responsibility for Gaza and Bethlehem. Yassin said Hamas went into the meeting with the intention of conducting serious discussions.

Palestinian Prime Minister Mahmoud Abbas and his Security Minister Mohammed Dahlan are due in Gaza today to meet with Albuhaeiry.

Handling the American side of the pressure was State Department official David Satterfield, who arrived yesterday with John Wolf, in charge of the American monitoring of the road map.

The American effort to keep the road map on track continues in Amman later this week when the four leaders of the international Quartet, which drew up the road map, will convene for an economic conference attended by among other U.S. Secretary of State Colin Powell.

The Palestinian cabinet convened last night to approve the security agreements hammered out between Israel and the PA over handing over control of the northern Gaza Strip to the PA.

At the end of the session, the Palestinian government issued a statement saying that "there is no such thing as one-sided implementation of commitments. The move to the political track and the implementation of the road map requires an end of the assassinations and Israeli incursions into Palestinian territories, as well as a lifting of the internal closures in PA areas and the siege of Yasser Arafat."

The process of preparing an IDF handover of northern Gaza to the Palestinian security services will continue in the next few days, Defense Minister Shaul Mofaz told the cabinet yesterday. The IDF is preparing to quit the Beit Hanun area and transfer it to the Palestinian Authority during the week. At the same time, Israel is giving positive consideration to a Palestinian request to expand the "pilot project' to include Bethlehem.

On Saturday night, Maj. Gen. Amos Gilad, the government coordinator in the territories, met at U.S. Ambassador Daniel Kurtzer's Herzliya residence with Palestinian Security Minster Mohammed Dahlan to discuss the arrangement. Israeli sources said the meeting was relatively positive and constructive.

Another meeting between Gilad and Dahlan, this time with Shin Bet officials and Palestinian counterparts is expected shortly. Israel expects the Palestinians to present a detailed plan for taking security responsibility for the northern Gaza Strip, with a particular emphasis on how it proposes to stop Qassam rocket fire from the area.

Yesterday, in addition to rockets fired from the Beit Hanun area toward Sderot, there were four rockets fired from the southern part of the Strip at Eshkol area settlements, an apparent Hamas shift to the southern part of the Strip.

Sources in the IDF say there is no reason for the Palestinians not to also get security control for Bethlehem. Lately there have been relative few terrorism alerts emanating from the Bethlehem area.

Mofaz said the Palestinians would be judged by the extent to which they prevent terrorism. "We are demanding that there be one address in every region and a mechanism for coordination at all levels," he said. He added the process will be "long, complicated and crisis-ridden."

"Israel will continue acting against `ticking bombs' in PA territories, after the U.S. approved it," Prime Minister Ariel Sharon told the cabinet. He explained there has been a change in the American position on the issue since last week's condemnation of the attempted assassination of Abdel Aziz Rantisi.

He said there are two reasons for this - the Americans understand that terror can foil the political process and the dimensions of the Israeli activity against terror. Military Intelligence chief Maj. Gen. Aharon Ze'evi also said "there is no legitimacy from the Americans to act against the Hamas - with balanced judgment, without hurting innocents, and without unreasonable collateral damage."

According to Sharon, "if there's no fire on us, we won't open fire, except for necessary operations for self-defense." He said the PA must accept responsibility for the areas under its control. Mofaz also said "if the PA does not take action against terror, we will."

Sharon's bureau chief Dov Weisglass is in Washington today for meetings with National security Advisor Condoleezza Rice, who is expected to tell him that Washington wants Israeli to hand over the security responsibility as quickly as possible. Washington sources said Rice and her team will present a tough line to Weisglass, demanding Israel renew its unqualified commitment to the road map.

Sharon meanwhile told the cabinet that Israel has prepared a set of principles for a meeting with Wolf, the American diplomat sent to oversee the implementation of the road map. The principles include the 14 reservations Israel's government attached to the road map when it approved two weeks ago. Wolf and his team arrived in the region yesterday.

President George Bush yesterday said he has not lost hope for peace in the Middle East but insisted the world must deal harshly with terrorists. Bush said he wants to help the Palestinians rein in terrorists, but he stopped short of pledging money or arms to the Palestinian Authority to aid it in combating terror groups like Hamas.

Pressed on what aid the Bush administration might provide, he said the U.S. is helping the Palestinian leadership complete a strategy for reconstituting its security forces "in order to make sure the terrorists, the haters of peace, those who can't stand freedom, do not have their way in the Middle East."

The remarks, as he left church in Kennebunkport, Maine, where he is spending a long father's Day weekend at his parent's summer home, were Bush's first on the Middle East since arriving in Maine.

Bush said he saw an apparent break in the surge of violence that followed a peace summit in Aqaba, Jordan, earlier this month. But he acknowledged the gravity of the job. "I'm confident we can achieve peace," he said. "It's going to be a tough road, but I am determined to continue to lend the weight of this government to advance peace."

"The way for us to have two states, side-by-side, is for everybody coming together to deny the killers the opportunity to destroy," Bush said. "And that's what they want to do. There are people in the Middle East who hate the thought of a peaceful Palestinian state. That's what they can't stand.

"It is clear that the free world, those who love freedom and peace must deal harshly with Hamas and the killers. And that's just the way it is in the Middle East."

Secretary of State Colin Powell and representatives of the United Nations, Russia and the European Union, which worked on the peace blueprint, plan talks in Jordan this week.