Militants are expected to attempt to disrupt next week's poll
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Iraqi authorities are distributing millions of copies of the proposed constitution to the electorate, a week before a referendum on the text.
The UN has urged Iraqi and US-led forces to ensure the document is available even in areas where fighting against militants is continuing.
President Jalal Talabani said most Sunnis taking part in the vote would reject the new constitution.
But he said it would not prevent its adoption by the country's electorate.
Sunni politicians have denounced the draft constitution as a recipe for the break-up of the country, and are mobilising their community to vote against it.
The US is trying to appease the Sunnis by urging Shia and Kurdish leaders to accept last-minute changes to the draft.
More violence
The UN, which is helping to organise the referendum on 15 October, is to distribute one copy of the constitution to every household when families collect their food rations.
The agency only received the final version of the text in the middle of last month, after representatives from the Shia and Kurdish political parties gave up trying to reach a consensus with the minority Sunnis.
One man picking up his copy of the constitution in Baghdad told the BBC's Caroline Hawley that he hoped the text would help end the violence.
As well as concern about distributing the text where fighting continues, there is concern about security at thousands of polling stations around the country, our correspondent says.
US army officials have repeatedly warned they are expecting a wave of attacks in the run-up to the vote.