The two-phased polling began amidst tight security and long queues were seen outside polling stations for the first round of vote in which a little more than half of Egypt's electorate of 51 million were eligible to vote.
A high turnout prompted the election organising committee to extend the voting till an hour before midnight here (0230 IST Sunday).
The second round of vote is scheduled for December 22.
Mursi and his Muslim Brotherhood have campaigned heavily in favour of the draft document, which may define Egypt for years to come. Opponents say it is poorly drafted and overtly favours Islamists.
People queued in long lines at polling stations in Cairo, Alexandria, Daqahliya, Gharbiya, Sharqiya, Assiut, Sohag, Aswan, North Sinai and South Sinai.
The run-up to the referendum saw deadly clashes between supports of the government and those against it. Soldiers joined police outside polling stations to secure referendum after deadly protests.
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The National Salvation Front, the umbrella opposition group, said it was deeply concerned about irregularities, such as directing voters outside polling stations to vote 'yes'.
In a statement, the group expressed "deep concern... Over the number of irregularities and violations in the holding of the referendum".
This, it alleged, pointed to a "clear desire for vote-rigging by the Muslim Brotherhood".
Judge Mohamed Abdel Hady, head of the media committee at the Judge's Club, that had refused to oversee the vote, said they had received hundreds of complaints, mainly regarding the absence of judges and the attempts to direct people to vote "yes" or "no". (MORE)