However, one prominent Sunni politician called the vote "a farce".
"Whatever the results of the referendum are... it is a civilised step that aims to put Iraq on the path of true democracy," said Farid Ayar, an official with the Independent Electoral Commission of Iraq.
He said the commission's 10-day audit of the vote had turned up no significant fraud and a UN official congratulated the commission for its work.
But Saleh al-Mutlaq, a Sunni Arab member of the committee that wrote the constitution, called the referendum "a farce" and accused government forces of stealing ballot boxes to reduce the percentage of "No" votes in several mostly Sunni-Arab provinces.
"The people were shocked to find out that their vote is worthless because of the major fraud that takes place in Iraq," he said on Al-Arabiya TV.
Many Kurds and a majority of Shi'ites strongly support the constitution, but Sunni Arabs fear it will create two autonomous and oil-rich mini-states of Kurds in the north and Shi'ites in the south, while leaving many Sunnis isolated in poor central and western regions with a weak central government in Baghdad.
The constitution is considered another major step in Iraq's democratic reforms, clearing the way for the election of a new, full-term Iraqi parliament on December 15.
Such steps are important in any decision about the future withdrawal of US-led forces from Iraq.
"The Iraqis are making inspiring progress toward building a democracy," US President George W Bush said after the result was announced.
"By any standard or precedent of history, Iraq has made incredible political progress, from tyranny to liberation to national elections to the ratification of a constitution in the space of two and a half years."
The militants, meanwhile, kept up their deadly attacks yesterday.
In the worst one, a suicide car bomb exploded near a regional government ministry in a predominantly Kurdish province of Sulaimaniyah, killing at least 12 people, said Dr Shirko Abdullaha at a local hospital.
It was one of two suicide attacks by insurgents in the generally peaceful province, which is 257 kilometres north-east of Baghdad.
In Baghdad, insurgents used four bombs and seven shootings yesterday to kill six people a boy, two Iraqi soldiers and three policemen and wounded 45 Iraqis, most of them policemen.
In Baqouba, 56km north-east of Baghdad, three insurgents riding in a car with hidden explosives failed to crash into a convoy of local government officials, crashing into a concrete barrier instead, killing two of them and wounding the other, officials said.
Meanwhile, Al-Qaida in Iraq claimed responsibility yesterday for the triple suicide bombings of Baghdad hotels on Monday that killed as many as 17 people.
The internet claim said the attack targeted the "dirty harbour of intelligence agents and private American, British and Australian security companies."