Al Jazeera's Mike Hanna in Cairo said early indications showed that the turnout had been low but that numbers were gathering as the day proceeded.
"This is a day that could change the course of Egyptian politics," he said.
The Muslim Brotherhood, the country's largest opposition force, said the Arab world's most populous nation faced a dark future if the changes win approval in the referendum.
"Everyone's being beaten over the head," Brotherhood leader Mohamed Mahdi Akef, told Reuters. "They've killed off everyone's hopes. Even those with a shred of hope had it killed off."
All major opposition groups including the Brotherhood, which won one-fifth of the seats in parliament in 2005 elections, have told their supporters not to take part because they have no confidence that the voting and the counting will be fair.
"It is really sad to see Egypt - a republic - turned into a kingdom"
Be Humble, UK
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Monitors say the turnout in Egyptian referendums has traditionally been low but the authorities tilt the balance by busing civil servants and public-sector workers to the polling stations on supervised voting trips.
The government says the changes are part of a gradual political reform process, but the opposition and human rights group say they are a step away from freedom and democracy.
The US joined in the criticism last week, saying it was concerned and disappointed that Egypt was not taking the lead in the Middle East on greater openness and pluralism.
But Condoleezza Rice, the US secretary of state, visiting Egypt on Sunday, tempered her criticism.
|"We recognise that states do this (political change) in their own way and that they do it in a way that is consistent with their own cultural circumstances"|
Condoleezza Rice, US secretary of state
"We recognise that states do this [political change] in their own way and that they do it in a way that is consistent with their own cultural circumstances," she said. "It is not a matter to try to dictate to Egypt how this unfolds."
The US campaigned for democratic change in Egypt in 2005, but analysts say Washington has since lost interest because it needs help from conservative Arab governments, such as Egypt, in Iraq and in the Israeli-Palestinian conflict.
Officials of the National Democratic party (NDP), the ruling part of Hosni Mubarak, the president, which has been in power in various guises for about 50 years, said they were aiming for a turnout of 25 per cent because of capacity limits at polling stations.
The authorities said the turnout at a similar constitutional referendum in 2005 was 54 per cent. Observers said the real figure was less than 10 per cent, possibly as low as five.
The amendments, which passed parliament one week ago after a secretive drafting process controlled by the NDP, target mainly the Brotherhood, an Islamist group which has built up strong support in Egyptian society since it was founded in 1928.
Under the amended constitution, Mubarak and the ruling party could dissolve the existing parliament and hold new elections under a new voting system which would make it more difficult for the Brotherhood to win seats.
|Mubarak is seen as trying to install his |
son as the next president [EPA]
The changes will enshrine in the constitution a ban on political activity or political parties with a religious basis or reference point.
Muslim Brotherhood leaders say the government's aim is to push the Islamists out and make it easier to install Mubarak's son, Gamal, as the country's next president.
Gamal Mubarak, who denies he has presidential ambitions, told a briefing on Sunday that he recognised the opposition.
"We are aware of the criticism and the scepticism," he said.
He denied that there was any link between the amendments and any plan to dissolve parliament.
"I don't see any reason for dissolving parliament," he said.