"The people of the region are looking towards Washington continuing to play a major role in the peace process by creating the necessary environment for a resumption of negotiations on the basis of the two state solution outlined in the Arab peace initiative and the Middle East roadmap," the palace statement said.
The Arab initiative, launched in 2002, would have traded diplomatic recognition for Israel's withdrawal from land it occupied during the 1967 Arab-Israeli war.
After Monday's summit with Mahmoud Abbas, the Palestinian president, and Ehud Olmert, the Israeli prime minister, Rice told a news conference that the two sides had reaffirmed their commitment to peace and previous agreements.
"All three of us affirmed our commitment to a two-state solution and agreed that a Palestinian state cannot be born of violence and terror," Rice said.
"I also want the Quartet to play a political role. I plan to sit and strategise with them about how we do that."
"The land of Israel belongs to both Jews and Palestinians - as long as there are people who reject this fact, the conflict will never be solved"
Iskander, Boston, USA
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The Quartet has demanded that the Palestinian government recognises Israel, renounce violence and accept previous peace accords in exchange for restoring foreign aid to the Palestinian territories.
Abbas argued on Monday that the unity government deal reached with the ruling Hamas party includes implicit recognition of these conditions and formally authorises him to negotiate with Israel.
The Palestinian president met King Abdullah on Tuesday before a trip to Europe to garner support for the unity government.
The king told Abbas: "Jordan respects the will of the Palestinian people, which was reflected in the agreement reached between Fatah and Hamas in Mecca.
"Jordan will step up diplomatic efforts in the coming phase to win international support to end the embargo imposed on the Palestinians and strengthen the position of the Palestinian negotiatior."
King Abdullah also spoke to Olmert by telephone calling on Israel to "contribute to reviving the peace process with the Palestinians and overcome all the obstacles".
Rice was also due to meet security and intelligence chiefs from the so-called Arab Quartet - Saudi Arabia, the United Arab Emirates, Egypt and Jordan - to discuss Israeli-Palestinian issues, a senior state department official said.
"We will be discussing the national unity government, what that means, particularly as that was brought in under Saudi sponsorship, and what we are trying to do with the trilateral," the official said before the meeting.
Speaking after the summit at a meeting with politicians from his Kadima Party, Olmert said he would maintain contact with Abbas, despite his dissatisfaction with the incoming unity government.
Olmert said: "We have to maintain a channel of communications with the Palestinians, and the only possible conduit is the Palestinian president."
It was the first meeting between Palestinian and Israeli leaders and a senior US official since June 2003.