Spurred on by a U.S. peace mission, Israeli troops pulled out of the Bethlehem area early Tuesday, edging closer to a cease-fire with the Palestinians in the 18-month-old Mideast conflict.

The pullback came after Vice President Dick Cheney arrived to bolster the efforts of U.S. mediator Anthony Zinni, and the pair held talks Monday with Israeli Prime Minister Ariel Sharon.

In the most promising sign since Zinni arrived last week, Israel pulled back after midnight from the West Bank town of Bethlehem — traditional birthplace of Jesus — and neighboring Beit Jalla, El-Khader and the Aida refugee camp.

In a statement, the Israeli military said its forces pulled out of the Bethlehem area and parts of the northern Gaza Strip.

The Palestinians demanded at security talks Monday afternoon that Israel pull out of all areas under complete Palestinian autonomy — about 17 percent of the West Bank — before a cease-fire could be declared.

Palestinian West Bank security chief Jibril Rajoub said the pullback from Bethlehem was not enough. He said Israel must complete its withdrawal from two other towns, Tulkarem and Qalqilya. Israel said it had pulled out of the two towns last week.

Also, Rajoub told The Associated Press that the Israelis must declare an "immediate end to their aggression in all its forms, including house demolition, closures, and assassination," and promise political negotiations as well as military talks.

The Bethlehem pullout ended one of the largest Israeli military operations in decades and takes Israel out of the more than a half-dozen Palestinian towns and cities it entered this month in a search for Palestinian militants. Earlier, they pulled back from the other West Bank Palestinian population centers.

"The meeting today was tough and serious, but positive," Rajoub said. "The Israelis are committed to withdrawing from all [Palestinian-run] areas in the West Bank."

It was not clear whether Israeli troops would remain in a few Palestinian-run areas of the Gaza Strip — a key road and some farmland — which they also seized in recent months, and this was apparently not part of the emerging deal.

In violence Monday, Israeli soldiers shot and killed an armed Palestinian near a crossing point between Gaza and Israel, the military said. Palestinians said he was a member of the militant Islamic Jihad. Later, soldiers killed two more Palestinians after an exchange of fire. The military said they were trying to plant a bomb.

A 52-year-old Palestinian was killed by tank fire when Israeli forces moved toward Dir al-Balah in central Gaza, Palestinian doctors said.

Palestinians fired two Qassam rockets into Israel from northern Gaza, the military said. Palestinians said Israel sent armored vehicles to search farms afterward.

Also, Israeli soldiers tracked down and arrested two Palestinians who had infiltrated into northern Israel, planning to carry out a terror attack, the military said.

Zinni, who has been shuttling between Israeli and Palestinian leaders, is trying to get the two sides to implement a cease-fire deal that CIA chief George Tenet brokered last year.

Both sides have previously endorsed the plan, which calls on the Israelis to pull back troops to the positions where they were before the fighting began in September 2000. The Palestinians must prevent attacks against Israel and collect weapons from militants.

Several previous cease-fire efforts have failed.

Many Palestinian militants say they only way they can win concessions from Israel is by fighting. And many Israelis believe Sharon should take an even tougher line and step up military operations against the Palestinians.

Both sides see Cheney's presence as an incentive to reach a truce deal.

Upon arriving, Cheney met Sharon and said he was seeking to revive Israeli-Palestinian talks with an aim to reaching a full-blown peace treaty based on U.N. resolutions. The vice president said both sides would have to make steps to end violence and improve the atmosphere for peace talks.

"We continue to call upon Chairman Arafat to live up to his commitment to renounce once and for all the use of violence as a political weapon and to observe a 100 percent effort to stamp out terrorists," Cheney said.

"In that same spirit, I will be talking to Prime Minister Sharon about the steps that Israel can take to alleviate the devastating economic hardships being experienced by innocent Palestinian men, women and children," he said.

Sharon, meanwhile, compared the U.S. campaign against international terrorism to Israel's battle against Palestinian attackers.

"The terrorists, their dispatchers and those who sponsor them must have no immunity or refuge, this is the only way to bring them to the realization that terror and violence will achieve nothing," he said.

The Palestinians, however, say they are resisting 35 years of Israeli military occupation in the West Bank and the Gaza Strip, with the aim of establishing an independent Palestinian state.

Rajoub said the occupation "is the worst kind of terrorism." In an interview published Monday in a Palestinian newspaper, Rajoub, considered a pragmatic moderate by many Israelis, would not criticize militias linked with Arafat's Fatah movement, though they have claimed responsibility for many recent bloody attacks against Israelis.

Cheney has set aside time to meet with the Palestinian side, aides said, although no specific meetings have been scheduled. A meeting with Arafat was a possibility, a senior U.S. official said on condition of anonymity.

Palestinian parliament speaker Ahmed Qureia said that unless Cheney meets Arafat he would not be received by any other Palestinian official.

"The address for an important person like Vice President Cheney is President Arafat," Qureia said.

Meanwhile, supporters of the radical Palestinian Islamic Jihad organization vowed to keep up attacks.

At a rally of about 600 people in the northern Gaza Strip town of Beit Lahia, demonstrators chanted "Death to Israel. Death to America," and '"No' to surrender, 'yes' to holy war."

"Our fighters will terrorize our enemy everywhere by all means," one masked activist told the crowd.

Since fighting erupted in September 2000, 1,210 people have been killed on the Palestinian side and 352 on the Israeli side.