Bush said he believed Palestinians would sign a treaty with Israel to establish their own state before he leaves office in about one year.
 
He said: "And I believe it's possible - not only possible, I believe it's going to happen - that there be a signed peace treaty by the time I leave office [in January 2009]."
 
Israeli pessimism
 
In stark contrast to the US president's glowing optimism, two public opinion surveys showed Israelis were overwhelmingly pessimistic over Bush's visit to the region. Most of them did not think the visit would help in moving forward peace negotiations.
 
The US president said he is convinced that both Israeli and Palestinian leaders understand "the importance of democratic states living side by side" in peace.
 
"I am confident that with proper help, the state of Palestine will emerge."
 
Abbas urged Bush to press Israel to halt Jewish settlements and ease security restrictions in the occupied West Bank that Palestinians say cripple their society and economy.
 
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Bad weather forced Bush to drive past settlements and the mammoth barrier Israel is building through the area, instead of travelling by helicopter.
 
"Our people did not forget your invitation and committment to the establishment of Palestine," Abbas told his US counterpart.
 
"Peace in the world starts from here, from the holy land."
 
Bush also urged Israel, which frequently mounts raids into the West Bank, not to take action that undermines Abbas's security forces.
 
He said: "There needs to be a fair amount of work to modernise the [Palestinian] security forces... my message to Israelis is that they ought to help, not hinder [them]."
 
On the issue of the Gaza Strip, which largely operates independent of Abbas's government,  Bush said: "There is a competing vision in Gaza."
 
He said the people of Gaza need to choose between Hamas, which has "delivered nothing but misery", and "those who have negotiated a peace settlement", meant to underscore his optimism over an agreement between Abbas and Ehud Olmert, the Israeli prime minister.
 
Hamas forced out Fatah and government forces from Gaza in June last year.
 
Palestinian resentment
 
In a move likely to draw Palestinian ire, Bush also said the two parties should leave behind unimplemented UN resolutions, such as those calling for the removal of Israeli settlements and a right of return for Palestinian refugees.
 

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He said: "The UN deal didn't work in the past... this is an opportunity to move forward and negotiate a new deal.
 
"We can stay stuck in the past, which will yield nothing good for the Palestinian people... or we can chart a hopeful path for the future."
 
David Chater, Al Jazeera's correspondent in Ramallah, said Bush put on an "appalling performance as far as the Palestinians here are concerned, they're not hearing anything they want to hear".
 
"He kept using, time and time again, the phrase 'the universality of freedom.' And he used it here, in Ramallah, that's been the centre of 40 years of military occupation. What universality of freedom do the people of Ramallah have, or anybody in the occupied West Bank?
 
"The economy is being choked off, the freedom is being choked off, the dignity of the people is also being choked off by what's happening under the occupation.
 
In Bethlehem, where the streets were sealed by Palestianian security forces, Al Jazeera's Jacky Rowland said residents felt "that their whole city has been put under curfew, effectively for a visit that's for them devoid of meaning".
 
After meeting Abbas, Bush was due to spend an hour touring the Church of Nativity before returning to Israel.
 
The Palestinian minister of tourism said she had tried unsuccessfully to persuade the US president to take a walking tour of Bethlehem - where the Israeli separation wall almost encircles the town, choking off its economy - in order to understand the situation there.
 
Rowland said: "There really is a question of what extent he's going to get any sense of what life is like for ordinary Palestinians here."
 
While in Ramallah, Chater added: "There's no sign at all from his two press conferences that Bush is even aware of the massive feelings of resentment in the occupied West Bank."
 
Outposts 'ought to go'
 
On Wednesday, the US president had begun his regional tour visiting Israel, and holding talks with Olmert.
 
Coinciding with renewed violence between the two sides, Bush told Israel to dismantle unauthorised settlement outposts and demanded that Palestinians halt rocket attacks.
  
Palestinian rockets were fired into Israeil on Wednesday, while an Israeli air strike killed three Palestinians in Gaza.

"I'm under no illusions," Bush said at a joint news conference with Olmert in Jerusalem. "It's going to be hard work."
 
Calling it "an historic moment and opportunity" for peace, Bush said Israel should remove Jewish settlements built without government authorisation in the occupied West Bank.
 
"Outposts, yeah, they ought to go," Bush said.
 
Olmert has pledged to remove them but has not given a deadline.
 
Peace groups estimate that there are currently more than 100 "wildcat outposts" in the West Bank - settlements that have not been approved by the Israeli government.