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.
The BBC's Andrew Harding witnessed the incident in Mbare - an MDC stronghold - and said the youths also threw punches at him and his film crew.
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Zimbabwe constitution: Key changes
Presidents limited to two five-year terms
Legislation can no longer be vetoed by the president
Devolves power to 10 provinces
Establishes peace and reconciliation commission
Creates independent prosecuting authority, anti-corruption commission, and guarantees independent electoral commission
Includes bill of rights which stipulates freedom of expression and a free media
Allows dual citizenship
Seizure of farms under the land reform programme can no longer be legally contested; president appoints land commission to carry out national audit
Although no-one was seriously injured, the incident is a reminder of the violence lurking close to the surface in Zimbabwe, our correspondent says.
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.'Vote in peace'
Polls across Zimbabwe opened at 05:00 GMT and are due to close at 17:00 GMT.
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.
Mr 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 Mr Gumbo as saying.
Analysts say the constitution is seen as a compromise document.
During campaigning the MDC has focussed on clauses guaranteeing protection against all forms of violence, and free and fair voting.
The MDC's Tendai Biti, who currently serves as finance minister, told the BBC the document was the "midwife" to a brand new Zimbabwe as it sets out people's rights, devolves some power and sets up a system of check and balances for those in authority.
The Zanu-PF campaign has highlighted the irreversibility of the land reform programme, which saw some 4,500 farms seized from mainly white commercial farmers, and other moves intended to give more economic power to black Zimbabweans.
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 organisations whose head office was raided by police in February, is deploying about 600 observers.
