His liberal rivals say it will deepen divisions in the nation.
Soldiers joined police outside polling stations to secure the vote after deadly protests. Demonstrations erupted last month after Morsi issued a decree expanding his powers and then fast-tracked the draft constitution through an assembly dominated by his Muslim Brotherhood and its allies.
"The sheikhs told us to say 'yes' and I have read the constitution and I liked it," said Adel Imam, a 53-year-old queuing to vote in a Cairo suburb. "The president's authorities are less than before. He can't be a dictator."
A coalition of leftists, socialists, Christians, who make up about 10% of Egypt's 83-million people, and more liberal-minded Muslims accuse Morsi of pushing through a document that does not reflect Egypt's diversity.
"I voted 'no' to the constitution out of patriotic duty. The constitution does not represent all Egyptians," said Michael Nour, a 45-year-old Christian school teacher in Alexandria, Egypt's second-biggest city.
Voters were allowed to cast "yes" or "no" ballots from 8am (0600 GMT) until 7pm (1700 GMT), but the deadline could be extended depending on turnout.
The vote has been split into two rounds, each covering different regions with the second next week, because not enough judges agreed to oversee the ballot.
There were queues of several dozen people outside some polling stations in Cairo and elsewhere as voting began. The first round covers about 26 million of Egypt's 51 million eligible voters. The next round is on December 22. – Reuters