are scheduled to close at 7:00pm in the first round. The rest of the country votes on December 22.
Alexandria, the country's second-largest city, was the scene of clashes on the eve of the referendum between opponents of the draft charter and Islamists after a cleric told worshippers at a mosque to support the constitution.
Hundreds of protesters later besieged the mosque, where the cleric took refuge.
The charter is at the heart of a power struggle between President Mohamed Morsi and the opposition, backed by judges who accuse the Islamists of overreaching.
The vote will be staggered over two rounds to ensure there will be enough judges to monitor polling stations amid a rift within the judiciary over the referendum process.
The first round's unofficial results are expected hours after the polling stations close.
Morsi has ordered Egypt's military to help police maintain security until the results of the referendum are known.
A total of 130,000 police and 120,000 soldiers are being deployed, interior ministry and military officials told AFP.
The main opposition coalition, the National Salvation Front, had mulled a boycott before urging Egyptians to vote down the charter, which rights groups say limits the freedoms of minorities and women.
International watchdogs, including the UN human rights chief, the United States and European Union, have expressed reservations about the draft because of loopholes that could be used to weaken human rights, including those of women, and the independence of the judiciary.
The referendum was only made possible after Morsi assumed sweeping powers that stripped a court of the right to annul the Islamist-dominated constituent assembly that drafted the charter, as some expected it to do earlier this month.
Morsi was forced to rescind his powers after mass protests outside his palace in northern Cairo led to the worst violence between the opposition and Islamists since his election in June.
Analysts said the proven ability of Morsi's Muslim Brotherhood movement to muster voters was likely -- but not certain -- to ensure that the draft constitution is passed.
The opposition, after failing to derail the referendum, has had little time to organise a rival campaign to scrap the charter, and has threatened to boycott the second round if it decides that the vote is unfair