BBC
The new constitution calls for a multi-party parliamentary election within three months.
The opposition boycotted the vote, calling it a farce and demanding President Bashar al-Assad stand down.
US Secretary of State Hillary Clinton has said "there's every possibility" Syria could descend into civil war.
But she warned "outside intervention" could exacerbate the situation.
"I think that as you try to play out every possible scenario, there are a lot of bad ones that we are trying to assess," she told the BBC.
Attempts to evacuate injured people, including foreign journalists, from the flash-point city of Homs were further delayed on Sunday.
'Laughable'
The Syrian government pressed ahead with organising the referendum despite the unrest, setting up more than 13,000 polling stations for 14.6m voters.
Among the voters casting his ballot in the capital was President Assad himself.
He sees the new constitution as the key element in a reform process he says will make Syria a beacon of democracy in the region, our correspondent says.
Under it, a multi-party system would replace the old monopoly of power enjoyed by the ruling Baath Party.
A year ago, that would have seemed revolutionary, our correspondent says, but activist and opposition groups have dismissed it as a sham, pointing out that the regime ignored many elements of the old constitution, which guaranteed personal and political freedoms and banned torture.
In many other parts of the country, voting was far from normal, our correspondent adds, with explosions and shooting reported from the east, west, north and south - in areas where violence has been going on for months.
At least nine civilians and four soldiers were killed in Homs, the UK-based activist group, Syrian Observatory for Human Rights, said.
Violent clashes were reported in the central city of Hama, the north-western province of Idlib and in Daraa province, south of Damascus.
Turkey's Foreign Minister Ahmet Davutoglu expressed doubt about the timing of the vote: "On one hand you say you are holding a referendum and on the other you are attacking with tank fire on civilian areas.
"You still think the people will go to a referendum the next day in the same city?" he said at a news conference in Istanbul.
The US has dismissed the referendum as "laughable".
'No progress'
"There are all kinds of negotiations, intensive negotiations, going on, and we hope to be able to do something as soon as possible, hopefully tomorrow," said Saleh Dabbakeh, an International Committee of the Red Cross spokesman.
Among those it is trying to help are two injured Western journalists, Edith Bouvier and Paul Conroy. It also wants to retrieve the bodies of another two journalists, Marie Colvin and Remi Ochlik, who were killed last week.
Also in the city are French photographer William Daniels and Spanish journalist Javier Espinosa.
A message posted from Espinosa's Twitter account, @javierespinosa2, on Sunday morning read: "Today less shelling and fighting but no trace of voting in bab al amr [Baba Amr]."
Hundreds of armed rebels from the Free Syrian Army are holding out in Baba Amr.
Meanwhile, international pressure is mounting on Mr Assad to end his government's 11-month crackdown on opponents.
Kofi Annan, the former UN secretary general who has been appointed the UN and Arab League's envoy to Syria, called for all parties to co-operate in finding a peaceful solution to the crisis.
The UN estimated in January that 5,400 people had been killed in the conflict. Activists say the death toll now is more than 7,300.
The Syrian regime restricts access to foreign journalists and casualty figures cannot be verified.
BBC's Jim Muir: "People have started voting... but there are also reports of shelling and shooting in many parts of the country"
The Syrian government has held a referendum on a new constitution as violence continues around the country, killing at least 30 people.
The opposition boycotted the vote, calling it a farce and demanding President Bashar al-Assad stand down.
US Secretary of State Hillary Clinton has said "there's every possibility" Syria could descend into civil war.
But she warned "outside intervention" could exacerbate the situation.
"I think that as you try to play out every possible scenario, there are a lot of bad ones that we are trying to assess," she told the BBC.
Attempts to evacuate injured people, including foreign journalists, from the flash-point city of Homs were further delayed on Sunday.
'Laughable'
The Syrian government pressed ahead with organising the referendum despite the unrest, setting up more than 13,000 polling stations for 14.6m voters.
Continue reading the main story 
The streets are filled with billboards encouraging people to vote for the constitution but the polling stations I visited were nearly empty.
At one school, two people showed up to vote in the 20 minutes I was there, although Ahmad Baalbaki, who was supervising the process, said 300 people had cast their vote.
Muneira, a housewife, says she voted "yes". She hasn't read the constitution but she supports the president.
In a central square, a pro-Assad rally is taking place in support of the constitution.
Nearby, a group of young men and women were handing out booklets to encourage people to vote.
One of them, Husein Saqer, said:"People don't want to vote because they see no difference between this one and the other constitution, but we are here to tell them to vote".
At the scene
At one school, two people showed up to vote in the 20 minutes I was there, although Ahmad Baalbaki, who was supervising the process, said 300 people had cast their vote.
Muneira, a housewife, says she voted "yes". She hasn't read the constitution but she supports the president.
In a central square, a pro-Assad rally is taking place in support of the constitution.
Nearby, a group of young men and women were handing out booklets to encourage people to vote.
One of them, Husein Saqer, said:"People don't want to vote because they see no difference between this one and the other constitution, but we are here to tell them to vote".
State television showed people voting in Damascus and various other places, and it all looked fairly normal, the BBC's Jim Muir reports from neighbouring Lebanon.
He sees the new constitution as the key element in a reform process he says will make Syria a beacon of democracy in the region, our correspondent says.
Under it, a multi-party system would replace the old monopoly of power enjoyed by the ruling Baath Party.
A year ago, that would have seemed revolutionary, our correspondent says, but activist and opposition groups have dismissed it as a sham, pointing out that the regime ignored many elements of the old constitution, which guaranteed personal and political freedoms and banned torture.
In many other parts of the country, voting was far from normal, our correspondent adds, with explosions and shooting reported from the east, west, north and south - in areas where violence has been going on for months.
Among the voters casting his ballot in the capital was President Assad himself
Violent clashes were reported in the central city of Hama, the north-western province of Idlib and in Daraa province, south of Damascus.
Turkey's Foreign Minister Ahmet Davutoglu expressed doubt about the timing of the vote: "On one hand you say you are holding a referendum and on the other you are attacking with tank fire on civilian areas.
"You still think the people will go to a referendum the next day in the same city?" he said at a news conference in Istanbul.
The US has dismissed the referendum as "laughable".
'No progress'
Continue reading the main story
“Start Quote
Message from trapped journalist Javier Espinosa's Twitter account @javierespinosa2Today less shelling and fighting but no trace of voting in bab al amr [Baba Amr]”
The International Red Cross and Syrian Arab Red Crescent were unable to enter the Homs suburb of Baba Amr on Sunday to evacuate injured Syrians and foreign journalists, but hope to try again on Sunday.
Among those it is trying to help are two injured Western journalists, Edith Bouvier and Paul Conroy. It also wants to retrieve the bodies of another two journalists, Marie Colvin and Remi Ochlik, who were killed last week.
Also in the city are French photographer William Daniels and Spanish journalist Javier Espinosa.
A message posted from Espinosa's Twitter account, @javierespinosa2, on Sunday morning read: "Today less shelling and fighting but no trace of voting in bab al amr [Baba Amr]."
Hillary Clinton: "I think there is every possibility of a civil war"
Meanwhile, international pressure is mounting on Mr Assad to end his government's 11-month crackdown on opponents.
Kofi Annan, the former UN secretary general who has been appointed the UN and Arab League's envoy to Syria, called for all parties to co-operate in finding a peaceful solution to the crisis.
The UN estimated in January that 5,400 people had been killed in the conflict. Activists say the death toll now is more than 7,300.
The Syrian regime restricts access to foreign journalists and casualty figures cannot be verified.
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