The vote will be staggered - many Egyptians will vote today but because there weren't enough judges to oversee the voting due to a boycott, other cities will vote on the constitution next week.
Either way, it is assured passage due to opposition parties urging their members not to take part.
Egyptians queued in long lines on Saturday to vote on a constitution promoted by its Islamist backers as the way out of a political crisis and rejected by opponents as a recipe for further divisions in the Arab world's biggest nation.
Soldiers joined police to secure the referendum after deadly protests during the buildup. Street brawls erupted again on Friday in Alexandria, Egypt's second city, but voting proceeded quietly there, with no reports of violence elsewhere.
President Mohamed Mursi provoked angry demonstrations when he issued a decree last month expanding his powers and then fast-tracked the draft constitution through an assembly dominated by his Muslim Brotherhood group and its allies. At least eight people were killed in clashes last week outside the presidential palace.
The liberal, secular and Christian opposition says the constitution is too Islamist and tramples on minority rights. Mursi's supporters say the charter is needed if progress is to be made towards democracy nearly two years after the fall of military-backed strongman Hosni Mubarak.
"The sheikhs (preachers) 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 country will move on."
Opposition politician and Nobel Peace Prize laureate Mohamed ElBaradei wrote on Twitter: "Adoption of (a) divisive draft constitution that violates universal values and freedoms is a sure way to institutionalize instability and turmoil."
Official results will not be announced until after a second round of voting next Saturday. But partial results and unofficial tallies are likely to emerge soon after the first round, giving some idea of the outcome.
Some observers have pointed out that this constitution isn't much more religious than the old one. That may be true, but the interpretation of the document will be in the hands of Islamists. You can bet that they will put the most religious spin as possible when it comes to enforcing the law.