Both main parties – Zanu-PF and the MDC – are urging their supporters to back the constitution, which would pave way for new elections later this year.
The polls could end a shaky power-sharing deal between the rival parties following a disputed vote in 2008.
Campaigning for the referendum was marred by an attack on an MDC politician in the capital, Harare.
Sten Zvorwadza, who hopes to become the next Movement for Democratic Change (MDC) lawmaker for the city’s Mbare suburb, was punched as he tried to put up posters on Friday.
Although no-one was seriously injured, the incident is a reminder of the violence lurking close to the surface in Zimbabwe.
On Tuesday, a Zanu-PF official was injured after his house was petrol-bombed by unknown assailants in Makoni district, in north-eastern Zimbabwe.
And in February, the 12-year-old son of an MDC activist died in an alleged arson attack in the eastern farming district of Headlands.
Polls across Zimbabwe opened at 07:00 local time and are due to close at 19:00.
In Mbare, the scene of Friday’s violence, more than 100 people were queuing outside a polling station as it opened, Reuters news agency reports.Under the new constitution, the president who wins the election, expected to be held in July, will be able to serve a maximum of two terms.Incumbent President Robert Mugabe, who has the backing of Zanu-PF, and MDC leader Morgan Tsvangirai, who is currently serving as prime minister, are both expected to compete for the presidency again.
Robert Mugabe, 89, has been in power since independence in 1980.
Zanu-PF spokesman Rugare Gumbo said the president wanted people to vote peacefully.
“He said we should not have violence and added that the party will not tolerate that nonsense whether it is intra-party or inter-party. He wants the people to vote in peace,” Zimbabwe’s state-run Herald newspaper quotes Rugare Gumbo as saying.
Analysts say the constitution is seen as a compromise document.
Western and US observers have been barred from monitoring the referendum, but some 2,000 local and other foreign observers have been accredited for Saturday’s vote, the Herald reports.
The Zimbabwe Election Support Network (ZESN), a network of 31 non-government organizations whose head office was raided by police in February, is deploying about 600 observers.
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