The results from Saturday's referendum are expected to show strong support for the text which would introduce presidential term limits, beef up Parliament's powers and set elections to decide whether the 89-year-old Mugabe stays in power.
Mugabe has ruled uninterrupted since the country's independence in 1980, despite a series of disputed and violent polls and a severe economic crash propelled by hyper-inflation.
The draft constitution is part of an internationally backed plan to get the country on track. Zimbabweans' verdict on the draft is expected to be known within five days of the voting.
Mugabe has backed the proposed constitution, which enshrines his drive to put land in the hands of black Zimbabweans. His political rival Prime Minister Morgan Tsvangirai has also lent his support to the text.
But that has not prevented the threat of violence from looming over the vote, as party militants keep one eye on the general election slated for July.
Shortly before polls opened on Saturday, gunmen later identified as plainclothes police detectives seized a member of Tsvangirai's Movement for Democratic Change (MDC) from his home north-east of Harare.
Police spokesperson Charity Charamba told Agence France-Presse Samson Magumura had been arrested on charges of attempted murder in connection with a recent firebomb attack that injured a Mugabe ally.
While casting his vote on Saturday, Mugabe, whom many blame for past unrest, urged Zimbabweans to ensure the referendum proceeded peacefully.
"You can't go about beating people on the streets, that's not allowed, we want peace in the country, peace, peace," he said.
Mugabe, the target of 11 years of Western sanctions over political violence and rights abuses, also used the opportunity to vow that United States and European monitors would not be allowed to observe the upcoming general election.
"The Europeans and the Americans have imposed sanctions on us and we keep them out in the same way they keep us out," he said.
Tsvangirai on Saturday expressed hope that a positive outcome would help catapult the country out of a crisis marked by bloodshed and economic meltdown.
He hoped the vote would move Zimbabwe "from a culture of impunity to a culture of constitutionalism".
Turnout, which was slow at first, picked up slightly as the day progressed, said Rita Makarau, chairperson of the Zimbabwe Electoral Commission, which has registered around six million eligible voters.
Minutes before the polls closed late on Saturday people in the Harare township of Mbare were still queing up to vote by candlelight.
Toilet rolls were improvised for candle holders as a gas lamp that was placed in the centre of a tent was insufficient to light the polling station.
Voters were handed a ballot paper in one hand and a candle in another before they disappeared behind a voting booth.
School teacher Petronella Dzikiti said she voted in favour of the new constitution, in part because it would introduce presidential term limits.
"We don't want a situation like we have today, where some of us knew one leader as a child who remains there when we are grown-ups," the 36-year-old said outside a polling station in Chitungwiza, near the capital.
Mugabe could rule until 99
The new constitution would for the first time put a definite, if distant, end date on Mugabe's rule.
Presidents would be allowed to serve two terms of five years each, meaning Mugabe could rule until 2023, when he would be 99.
The text would also strip away presidential immunity after leaving office, bolster the power of the courts, and set up a peace and reconciliation commission tasked with post-conflict justice and healing.
In the run-up to the vote, violence did not approach the levels seen in the disputed 2008 elections.
Then, at least 180 people were killed and 9 000 injured in a crisis that ultimately forced Mugabe and Tsvangirai into a power-sharing government.
There are widespread fears that July's election might bring a return to bloodshed.
On the eve of the referendum, several MDC members, including a parliamentary candidate, were beaten up as they put up posters backing the draft constitution.
The authorities have also been accused of targeting pro-democracy groups by arresting their leaders and seizing equipment.
Tsvangirai called on SADC leaders to meet for an urgent summit, to help ensure the election is fair and free of violence and intimidation.
Observers fear there may not be enough time to apply all the necessary reforms to ensure a healthier political environment before the next elections. – AFP