Story Highlights• Condoleezza Rice meets with Israeli and Palestinian leaders
• Rice vows more talks as she tries to jump-start Mideast peace process
• The position of new Palestinian unity government is at issue
• U.S., European leaders demand recognition of Israel, opposition to terrorism
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JERUSALEM (CNN) -- U.S. Secretary of State Condoleezza Rice held "productive" talks Monday with Israeli and Palestinian leaders and said she expects more negotiations in a bid to restart the Mideast peace process.
Rice's meeting Monday with Israeli Prime Minister Ehud Olmert and Palestinian Authority President Mahmoud Abbas came amid concerns about whether a new Palestinian unity government will renounce terrorism and recognize Israel. (Watch why Rice's visit comes at a difficult time )
Rice issued a statement afterward calling the talks "useful and productive."
"All three of us affirmed our commitment to a two-state solution, agreed that a Palestinian state cannot be born of violence and terror, and reiterated our acceptance of previous agreements and obligations, including the road map," Rice said, referring to the 2003 internationally backed plan for peace.
Rice offered no specifics.
Miri Eisen, Olmert's spokeswoman, said there had been a "lowering of expectations for the outcome" of the talks. "The importance of this meeting is in their sitting down and having dialogue," she said.
"We try to define it as confidence-building. You have to meet somebody and talk to him and see if you have confidence and trust each other and see if there is a sense that you can go forward."
The meeting of Israeli and Palestinian leaders came despite no sign that a new Palestinian government will do what world leaders have called for: renounce terrorism and recognize Israel.
Fatah and Hamas are working to build a unity government, but tension and fighting between the two factions have left the future of the Palestinian government unclear.
The so-called Mideast Quartet -- the United States, European Union, Russia and United Nations -- has called on the new government to renounce terror and recognize Israel, making those steps the keys for the resumption of governmental aid.
Abbas and other leaders of the Fatah party support those steps, but Hamas, which has controlled the government for the past year, is staunchly against them. Hamas' victory, and its refusal to change its stance, led the United States and European Union to cut off millions of dollars in foreign aid.
"I haven't seen anything to date that suggests this is a government that is going to meet those quartet principles, but we will see when that government is formed," Rice said Sunday. "Let's see how President Abbas deals with the new circumstances."
Rice met separately Sunday with Abbas and Olmert.
U.S. government officials do not meet with leaders of Hamas, which has carried out terrorist attacks and is listed by the United States and Israel as a terrorist organization.
"I think that eventually, one way or another, there is going to have to be a Palestinian government that accepts the quartet principles, because that is the only basis on which you can get a state," Rice told reporters. "It simply can't be the case that the political horizon can be built on the basis that one of the parties doesn't accept the other's right to exist."
Rice's meeting with Abbas in the West Bank town of Ramallah lasted more than two hours, most of it without the presence of aides. (Watch Rice struggle to breathe life into "complicated" peace deal )
Addressing reporters, Rice thanked Abbas for his "personal commitment to peace and understanding, which has led the Palestinian people to international agreements that recognize the importance of the two-state solution and all that comes along with that."
A key goal is to get aid dollars flowing again from the United States and the European Union. Both have continued to channel money through nongovernmental organizations to assist Palestinians, but the Palestinian government has been crippled financially.
The government already faced severe financial difficulties under Fatah. Hamas won last year's elections largely on a promise to clean up Fatah's corruption.
Rice indicated that the Israeli-Palestinian conflict, which exacerbates tensions throughout the region, is a key priority during the remains years of the Bush administration.
"I am committed to this," she said. "It takes patience, it takes perseverance, it takes getting up after a bad day and trying to make a better day, and that's what I'm going to do as long as I am secretary of state."
CNN's Elise Labott contributed to this report.
U.S. Secretary of State Condoleezza Rice meets Monday with Palestinian leader Mahmoud Abbas, left, and Israeli Prime Minister Ehud Olmert in Jerusalem.