Interior Minister Habib el-Adli said 53.6% of registered voters turned out for Thursday's referendum, which opposition groups had asked Egyptians to boycott.
The Egyptian government later said the "yes" vote was a "great success for democracy under the leadership of President Hosni Mubarak".
The new system, replacing referendums on a single presidential candidate chosen by a parliament dominated by the ruling party, takes effect in elections in September.
President Mubarak, 77, who has run the country since 1981, is expected to seek a fifth six-year term but has not yet said whether he will stand.
The opposition says the conditions on presidential candidates are so restrictive that the ruling party would not face a credible challenge.
An opposition spokeswoman said the government had exaggerated the turnout in the referendum.
Journalists saw no voter queues
before polling stations
"This rate of attendance is just impossibly high ... . It seems the number has been multiplied by five," Gameela Ismail, spokeswoman for the opposition Ghad (Tomorrow) Party, said.
Mohammed Habib, deputy leader of the Muslim Brotherhood, which joined the boycott call, said: "I don't believe this number is possible. We know not more than 5 or 10% of people go to vote ... . I have many doubts about this number."
Abdel Halim Qandil, a leading member of the Kifaya (Enough), protest movement said the real turnout was about 4%.
Some journalists covering the referendum saw no queues before polling stations and only a trickle of voters.
But in many places government agencies and companies bused their employees to polling stations to encourage voting.
The referendum was also marred by attacks by supporters of the ruling National Democratic Party (NDP) on Kifaya members who demonstrated in favour of a boycott.
"This rate of attendance is just impossibly
high ... . It seems the number has been multiplied by five"
Al-Ghad Party spokeswoman
Riot police stood by as Mubarak's supporters kicked and punched a small group of people from Kifaya, an alliance of liberals, leftists and Islamists dedicated to getting Mubarak out of office and prevent his son Gamal replacing him.
The men groped women and ripped their clothes.
The US condemned the assault on the protesters.
"There is no excuse for attacks on peaceful demonstrators," White House spokesman Scott McClellan said. "Anyone who attacks peaceful demonstrators must be arrested and tried, that's our view."