The constitution has split the country and caused tempers to flare
Egyptians are voting in a referendum on a draft constitution that has divided the country and sparked deadly unrest.
President Mohammed Morsi and his Muslim Brotherhood have campaigned heavily in favour of the constitution.
Opponents say it is poorly drafted and too Islamist.
This round of voting takes place in Cairo, Alexandria and eight other provinces, a week before the rest of the country. Some 250,000 security personnel have been deployed.
More than 51 million people are registered to vote.
Polling had to be spread out because few judges were willing to supervise the referendum.
Human rights groups have expressed fears that results from the first round could sway the opinion of those voting in the second.
The referendum asks Egyptians whether to accept or reject a basic document that must be in place before elections can be held early next year.
The BBC's Jon Leyne in Cairo says opponents of the constitution argue it leans too heavily towards the imposition of Islamic law - and that supporters like it for exactly that reason.
He says the referendum is more than a vote on obscure clauses; it is about the future direction of Egypt - whether it should be an Islamic country or a secular one.
"The times of silence are over," said Essam el-Guindy, a bank worker queuing to vote in the Zamalek district of Cairo.
"I am not OK with the constitution. Morsi should not have let the country split like this," he told
But 53-year-old voter Adel Imam told Reuters: "The sheikhs told us to say 'yes' and I have read the constitution and I liked it."
"The president's authorities are less than before. He can't be a dictator."
Both sides brought out supporters for final rallies on Friday.
Clashes flared in the northern port city of Alexandria, where rival activists fought with clubs, stones and other weapons. A number of cars were set alight and at least 15 people were injured.
The violence reportedly broke out after a cleric at a mosque had urged worshippers to vote "Yes".
Witnesses reported fresh clashes in the city late on Friday, with police firing tear gas.
Security is expected to be intense on Saturday, with President Morsi granting the army the power to arrest civilians.
The opposition National Salvation Front (NSF) coalition had vehemently opposed the referendum but this week said its supporters should go to the ballot boxes to vote "No".
Ahmed Said, leader of the Free Egyptians Party, a part of the NSF, said: "History will remember that this regime forced a referendum on the people of Egypt in these harsh circumstances.
"They can't find judges to monitor and [there is] blood on the streets."
In half-page advertisements in newspapers on Friday, the opposition called the document "a constitution that divides Egypt".
But supporters of the draft constitution accused the opposition of sowing "lies and discord" about the referendum.
Amr Darrag, of the Muslim Brotherhood's political arm, the Freedom and Justice Party, said: "This is political blackmail that is not based on any evidence or reality."
Our correspondent says that even if one side or the other wins a clear victory, it is not likely to be the end of a debate that has divided Egypt down the middle.