Bush, speaking to reporters before their talks in the Oval Office, said Olmert kept his word - "and in international politics, that's important."
"We've been through a lot together during our time in office," Bush said. "We strongly believe that Israel will benefit by having a Palestinian state, a democracy on her border that works for peace. ... It's not easy to try to change the paradigm, and I understand that."
"I just want you to know that I believe that vision is alive and needs to be worked on."
Olmert said history will owe Bush a debt of gratitude for setting the region on the path to an accord.
"A two-state solution is the only possible way to resolve the conflict in the Middle East," Olmert said.
"I'm sure that when the history books will be written, the contributions that you made to the safety and security of many people will be greatly appreciated."
Neither Bush nor Olmert, however, can put an Israeli-Palestinian peace accord in their legacy book.
Just a year ago, at a summit Bush hosted in Annapolis, Md., Olmert and the Palestinian president, Mahmoud Abbas, resumed peace talks after a seven-year standstill. The three set an ambitious target to have a final peace deal by the end of 2008. But despite a lot of talk, Bush's two trips to the region and eight more by Secretary of State Condoleezza Rice, all have acknowledged the year-end target will not be met.
Bush administration officials say some progress has been made.
"We're much farther along the road than we otherwise would have been" without the Annapolis meeting, White House deputy press secretary Tony Fratto said Monday. "And I think we're much farther down the road than most people give the Palestinians and the Israelis credit for.
"Now, we do have the elections coming up in Israel, and obviously that will delay or throw off the process a bit."
The ongoing threat from Iran's nuclear program also was a central topic of their discussion.
Israel sees it as the biggest national security threat, and also has been provoked by Iranian President Mahmoud Ahmadinejad's repeated calls for the destruction of the Jewish state.
The U.N. Security Council has imposed three rounds of economic penalties on Iran, which insists its nuclear program is peaceful and designed to produce energy. Both the U.S. and Israel say they hope diplomatic pressure resolves the standoff, but have not ruled out military action.
A report this past week by the International Atomic Energy Agency said Iran was stonewalling attempts to monitor its nuclear activities. And intelligence assessments from Israel, obtained Monday by The Associated Press, said weakening international pressure on Iran will embolden the government in Tehran to make major strides next year toward developing a nuclear bomb, something Israel thinks Iran will be capable of building by 2010.
Earlier Monday, the Israeli leader had breakfast with Rice and national security adviser Stephen Hadley and met with Vice President Dick Cheney. Bush, Olmert, first lady Laura Bush and the Israeli prime minister's wife, Aliza, dined privately at the White House.
