By Paul Wood
BBC News, southern Iraq
Voting was quiet, calm, and steady at the polling station we visited just south of Basra.
In southern Iraq, Shias welcome the constitution
There was not the excitement of January's election, but there was still an atmosphere of celebration among Iraq's Shias as they waited to vote.
One man said with a big grin that he was very happy to be able to take part in the referendum - only the second time in decades that Iraqis have been able to cast a democratic ballot.
By mid-morning, turn-out at this one polling station was already high: 2,000 out of 2,900 registered voters.
Almost everyone we spoke to was voting "Yes" but people are doing so for different reasons.
Religious Shias were mindful of the instruction from senior clergy that supporting the constitution was a religious duty.
Some admitted they had not even read the document they were voting for: its endorsement by the Grand Ayatollah Ali Sistani was enough.
Civil war fears
We spoke to a secular Shia - a university lecturer - who had misgivings about the new constitution but said it would be unbearable for Iraq to turn the clock back a year and start the negotiations again.
It could lead to civil war, he said, and people needed political stability to start making simple improvements in their lives.
That was an opinion we heard a lot from voters as concerned about power, water and security as about high politics.
A gaggle of women in black abayas emerged from the polling room. Each had a forefinger stained with ink to show they had voted.
Both Iraqis, and the British know, that the closer Iraq gets to political stability, the closer the international forces will be to going home
All had voted "Yes" except for one woman who said, without further explanation, that the constitution was not democratic enough and she had voted against.
One man, wearing the flowing tan and gold robes of a Marsh Arabs Sheikh said: "I am voting 'Yes' - to end the occupation."
Many Iraqis, on both sides of the sectarian divide, share that opinion, too.
Outside, British troops waited a respectful distance from the polling station.
The coalition was anxious not to have even the appearance of trying to influence an Iraqi democratic process.
But in Basra itself, relations with the police remain tense after last month's trouble and British troops stayed at least a mile (1.6km) away from the nearest polling station.
Both Iraqis, and the British know, that the closer Iraq gets to political stability, the closer the international forces will be to going home.