At least eight people, among them three policemen and a US soldier, were killed in attacks across the country on Thursday, but tight security may be working. A US general said daily attacks were down some 40% compared to January's election campaign.
Shia and Kurds - some three quarters of the 15 million voters - are likely to ensure a "Yes" majority nationwide, but a two thirds "No" vote in three of 18 provinces would veto it.
Though many in the Sunni regions of the west and north oppose the constitution, division among their leaders and threats against voters make a veto unlikely. That said, straw polls around the country show widespread mixed feelings.
Announcing nationwide curfews as the first Iraqis cast early ballots in hospitals and prisons, Interior Minister Bayan Jabor said frontiers would close from midnight (2100 GMT) until Sunday. Businesses shut for a four-day holiday and private vehicles were banned from overnight travel and all day Saturday.
Two attacks damaged a Sunni group's offices in Falluja after the Iraqi Islamic Party broke ranks and backed the constitution once parliament had made some minor amendments on Wednesday.
Private vehicles will be banned
all day on Saturday
Washington was concerned that though defeat for the charter was unlikely, forcing it through in the teeth of Sunni hostility could heighten rather than lessen the risks of further violence.
US Ambassador Zalmay Khalilzad, credited with "tireless" mediating efforts by Iraq's president, conceded there was still much negotiation ahead to bridge the remaining divides.
"The draft constitution ... is moving toward becoming the national compact Iraq needs," he told reporters. "There are issues that the Iraqis have to deal and come to terms with, like federalism, the role of religion, resources, rights of women."
At Baghdad's Yarmouk hospital, often the theatre for the human misery left by attacks in the capital, doctors as well as patients were able to vote on Thursday.
"People feel good about this new change and about the deal the political parties reached," said Hussein, a member of staff.