President Bush didn't give any final vote count on Sunday, but called Saturday's vote on the constitution in Iraq and the high turnout good news.

Returning to Washington from Camp David (search), Bush stopped by reporters' cameras to congratulate the people of Iraq and characterize the vote as a victory for opponents of terrorism.

"This is a very positive day for the Iraqis and, as well, for world peace. Democracies are peaceful countries. The vote today in Iraq stands in stark contrast to the attitudes and philosophy and strategy of Al Qaeda (search) and its terrorist friends and killers. We believe and the Iraqis believe the best way forward is through the Democratic process," the president said.

Bush said Sunni participation added to the turnout, which was expected to exceed that from the election earlier this year when the interim Iraqi National Assembly (search) was approved. The president noted less violence during this vote despite threats from terrorists to disrupt the vote.

"That's a tribute to the Iraqis -- forces who we've trained, as well as coalition forces that worked hard to make sure that democracy could move forward in Iraq," Bush said.

The United States is hoping the constitution, if approved, will build confidence in the new government among minority Sunni Arabs and sap support for the insurgency. Some Sunni Arab leaders agreed late last week to encourage backers to support the constitution, which was developed after months of difficult negotiations.

Sunnis are still concerned, however, with power-sharing measures given to Shiites and Kurds. They claim the agreement only papers over the cleavages in a religiously, ethnically and culturally mixed society.

The president grabbed the opportunity to speak to what he said is an important step for the Middle Eastern nation. Political stability could also lead to the draw down of some of the 150,000 U.S. troops stationed in the nation.

In London, Secretary of State Condoleezza Rice (search) said that violence will continue in Iraq even if the new constitution is approved. She said that support for the insurgency will wane as the country moves toward democracy.

"There's no doubt that a few violent people can make life really miserable for the majority of the Iraqi people. ... But they are not stopping the political process. And that's bad for the insurgency, because you defeat an insurgency not just militarily but politically as well. And clearly, the Iraqi people now believe that their future rests with this political process. There is no political base any longer for this insurgency," Rice told "FOX News Sunday."

Initial estimates showed a high overall turnout of about 61 percent and reports of no major attacks, spokesman Allen Abney said Saturday.

Insurgents attacked five of Baghdad's 1,200 polling stations, but no one was killed. The only deaths reported were of three Iraqi soldiers hit by a roadside bomb far from a polling site. Separately, five U.S. soldiers were killed by a roadside bomb Saturday while patrolling during voting in Ramadi, west of Baghdad.

Amid a surprisingly large turnout in several key heavily Sunni provinces, many voters said they were voting against the constitution. If two-thirds of voters in any three of Iraq's 18 provinces vote "no," the charter will not be approved.

One Sunni leader estimated on Sunday that 80 percent of that ethnic group voted against the constitution. But rejection of the constitution appeared highly unlikely after initial vote counts showed that a majority supported the constitution in two of the four provinces Sunni Arab opponents were relying on to defeat it.

Rice at first predicted the document would pass the referendum, but later backed off. She told "FOX News Sunday" that the administration still doesn't know "the fate of the referendum at this point. They're still trying to do the kind of exit analysis that would give them a sense of what areas voted in what numbers," Rice said.

"But whatever happens with the referendum, the important matter is that the Iraqis have in large numbers gone out to vote in this process," she said.

If adopted, the draft charter would provide the basis for a general election in two months for a full-term parliament. Results were not expected for a few days. If the constitution fails, a new one must be drafted by a new parliament, to be elected in December.

The U.S. views success in the election as critical to undermining the insurgency and being able to begin phasing out the U.S. military presence in Iraq that is increasingly unpopular with the American public.

Less than 40 percent in an AP-Ipsos poll taken in October said they approved of the way Bush was handling Iraq. Just over half of the public now say the Iraq war was a mistake.

Sen. Carl Levin, D-Mich., ranking minority on the Senate Armed Services Committee, told a network news show on Sunday morning that the Iraqi constitution is a divisive document that even if approved leaves key political issues unsettled, such as autonomy for certain regions.

"That means the political unity, which is absolutely essential to defeat the insurgency, does not exist in Iraq," he said. Still, Levin said he expected tens of thousands of American troops to return home before the November congressional elections next year.

Democratic Sen. Dick Durbin (search) of Illinois told "FOX News Sunday" that if the turnout is one of rejection, "we have to go back to the drawing board, start all over on the constitution."

Durbin also criticized the administration for not coming up with a plan for the peace after deposed Iraqi President Saddam Hussein (search) was ousted.

"Our men and women in uniform have not failed. The political leaders have failed, failed to come up with a plan which said after Saddam Hussein is gone, this is how we will end this war," he said. "This administration has a responsibility to the American people to do better. America can do better. Our soldiers deserve better than what we've received from this administration."

To that, Republican National Committee Chairman Ken Mehlman (search) criticized Durbin for what he said was a negative outlook on a positive event.

"One day after 9 million Iraqis went to the polls to vote on a new constitution, Senator Durbin returned to his partisan message of pessimism and defeat, ignoring their progress while offering no alternative. Americans can do better than political leaders whose desire to score partisan points blinds them to reality and prevents them from developing constructive alternatives," he said.

Separately, House Speaker Dennis Hastert (search), R-Ill., congratulated Iraqis in a written statement.

"Today's peaceful vote demonstrates a marked change in Iraq from the days of Saddam's torture chambers and one that should be a message to those who advocate withdrawing our troops," he said.

Added Senate Majority Leader Bill Frist, R-Tenn.: "While we can expect the terrorists to continue their attacks, they will not derail Iraq's continued political progress. I believe the success of the political process is the basis by which all Iraqis will eventually rise up in opposition to the terrorists and their dark view of the world."

FOX News' Julie Kirtz and The Associated Press contributed to this report.