BAGHDAD, Iraq (CNN) -- By nightfall Saturday election workers in Iraq were hand-counting the millions of paper ballots cast in the war-weary nation's constitutional referendum.The process, which involved more than 5,800 polling stations, is expected to take days.Although results aren't expected until next week, the referendum already was being hailed as a success, because turnout appeared to be high enough to legitimize the outcome -- and no major violence was reported. (Watch how people voted -- 2:33)"The success in this referendum, it isn't how many people are going to say 'yes' and how many people are going to say 'no,' " said Fareed Ayar, a spokesman for Iraq's Independent Electoral Commission. "The success is that all Iraqis ... found out that the polling station is the way to deal with the political problems in Iraq, to deal with the violence in Iraq.""That is the message today that the Iraqis are sending to the rest of the world," he said.About 15.5 million of Iraq's 26 million people were registered to vote. (A look at what's at stake -- 3:08)Initial figures showed more than two-thirds of eligible voters cast ballots in Baghdad and seven other provinces, said Ayar. In eight others, turnout ranged between 33 percent and 66 percent, he said.Electoral officials had no information about Anbar province in western Iraq, which has been a hotbed of insurgent violence. Turnout in the southern province of Qadisiya was projected to be less than 33 percent, Ayar said. The United Nations' top elections official, Carina Perelli, described voting as having gone "steadily" in all regions of the country. But she cautioned that initial turnout numbers were based on best guesses and unscientific counting and didn't take into account the possibility of irregularities.In January, when Iraqis elected an interim national assembly, about 60 percent of the registered voters turned out nationwide. Sunni Arab leaders actively boycotted that vote, then found themselves with little voice in government. By contrast, strong participation was reported Saturday in some of the Sunni Arab areas where voters were scarce in January.No major violenceElaborate precautions taken by Iraqi and U.S.-led coalition forces prevented any major incidents. Iraqi soldiers and police protected voting sites, while U.S. and coalition forces were on stand-by in case of trouble.Vehicles were barred from the streets, so voters walked to the polls. (Watch last-minute preparations -- 2:24There was scattered violence, however. Iraqi police said a sniper killed a civilian at a western Baghdad polling station. The sniper, who may have been targeting police, was not captured. An Iraqi police patrol near a polling station in Baghdad was hit by a roadside bomb shortly after voting started. Two Iraqi police officers were wounded, according to a police official.And 11 gunmen stormed a Baghdad polling place a half hour after polls closed at 5 p.m. (10 a.m. ET), making off with five boxes of ballots and wounding an electoral commission employee, police said. How many ballots were stolen was not clear.Election day was heralded Friday night when insurgents attacked a main power line into Baghdad, knocking out electricity to about 70 percent of the capital. By morning, power and water were being restored. (Full story)Reaction to the voteA White House spokesman said President Bush -- who has seen public support for his Iraq policy waver in recent months -- was "pleased to hear that Iraqis turned out in large numbers to freely express their views on this historic day.""Today's vote deals a severe blow to the ambitions of the terrorists and sends a clear message to the world that the people of Iraq will decide the future of their country through peaceful elections, not violent insurgency," spokesman Allen Abney said.A spokesman for U.N. Secretary-General Kofi Annan called Saturday's vote "an important opportunity for the Iraqi people to express their political views.""Whatever the outcome, the secretary-general believes that this referendum offers an opportunity for all Iraqis to move away from violence and to unite in a spirit of national reconciliation to build a democratic, unified and prosperous Iraq," the U.N. spokesman said.Sunnis' pivotal voteChances for approval of the constitution increased considerably Wednesday when the Iraqi Islamic Party -- the largest Sunni Arab party -- dropped its opposition after the transitional assembly agreed to consider changes in the framework once a general election is held in December.Sunni Arab groups have objected to provisions that would grant more autonomy to Shiite areas in the south and Kurdish areas in the north. They also object to provisions that exclude elements of former dictator Saddam Hussein's Sunni-dominated Baath Party.How the vote must goThe draft constitution -- hammered out after months of contentious, painstaking negotiations by lawmakers in Iraq's transitional National Assembly -- must be approved by a majority of Iraq's voters.With strong support in the Shiite and Kurdish communities, which together account for more than three-quarters of the population, that threshold is expected to be met.However, the constitution will fail if it was rejected by at least two-thirds of the voters in at least three of the country's 18 provinces. With many Sunni Arab groups opposing the document, rejection is considered possible in four provinces where Sunnis predominate. (Full story)Rejection of the constitution would be a serious blow to Iraq's political evolution since the U.S.-led invasion that toppled Saddam in 2003. The transitional assembly would be dissolved and the process of writing a constitution would have to start over after a new assembly is elected in December.By contrast, if the constitution is approved, Iraqis would vote in December for a new, permanent government -- possibly clearing the way for the United States and its coalition allies to begin withdrawing troops from Iraq.CNN correspondent Aneesh Raman, Arab Affairs editor Octavia Nasr and producers Arwa Damon and Mohammed Tawfeeq contributed to this report.