The two leaders, meeting in the Oval Office, said Abbas is a moderate voice and the only true leader of the Palestinian people. They also agreed, as Bush said, to work on a "common cause to fight off these extremists" in Gaza, where the Hamas terror group, led by deposed Palestinian Prime Minister Ismail Haniyeh, is basing its leadership.
Abbas is the only true "voice for moderation" Bush said, adding that he is the elected leader of the 1.4 million Palestinians in Gaza as well as the 1.5 million Palestinians in the West Bank.
"He is the president of all the Palestinians. He has spoken out for moderation. He is the voice, the reasonable voice among extremists in your neighborhood," Bush said.
"I'm going to make every possible effort to cooperate with him," the prime minister said of the Fatah Party chief who is also known by the nom de guerre Abu Mazen.
Violence in the Palestinian territories continued Monday as fast-moving changes over the past week left Abbas dissolving the Palestinian government and creating an emergency cabinet, led by Western-backed economist Salem Fayyad.
In response, the United States and European Union agreed to lift economic sanctions and send millions of dollars to help Abbas prop up his leadership despite his lack of control over the coastal Gaza strip.
Saying he wants to make sure security is in place to help all Palestinians, including those in Gaza, Olmert called Hamas' actions as "atrocious and intolerable" as he described how militants killed rivals by dropping them from the fifth story of buildings and going to a hospital to shoot them in their beds.
Bush said the split in the Palestinian territories is purely the fault of Hamas. "It was Hamas that attacked the unity government," the president said. "They made a choice of violence. It was their decision that has caused there to be this current situation in the Middle East."
Olmert said he wants to resume the biweekly talks he initiated with Abbas and would like to develop a security plan for the Palestinians. Though Olmert did not give any indications, Israel could free up millions in tax revenue it collects on behalf of the Palestinians to give to the Palestinian Authority.
Olmert added that the Palestinians must now work to fight terror, something he said Abbas has not done until now. Both Bush and Olmert said they are looking for opportunities for peace in the region in the framework of a two-state solution of Palestinians and Israelis living side-by-side.
"This is a very special time. Things happened lately, very dramatic," he said. "We, in the Middle East, are somewhat less surprised but not less outraged by such events. ... What we ought to do is try to find opportunities for the future. ... I want to strengthen the moderates and cooperate with Abu Mazen, who is the president of all the Palestinian people."
The prime minister also said he wanted to discuss with Bush the threat to Israel from Iran, whose president has said Israel must be "wiped off the map."
Bush replied that he views Iran's statements as a "serious threat" to Israel and that "all options are on the table" to make sure Iran does not develop a nuclear weapon. Bush said Iran must see that there is "a price to be paid for this kind of intransigence."
The Associated Press contributed to this report.