The figures were provisional and had yet to be confirmed, the interior ministry cautioned.
Turnout was between 65 and 73 percent, but the final figure was not expected until later.
Voting ended at 7:00 pm (GMT and local time), and took place "under good conditions, without incident or problem," said interior ministry spokesman Sidi Yeslem Ould Amar Cheine.
The government was organising a special electoral results evening as figures trickled in from 2,329 polling stations scattered across an area twice the size of France.
Final official results were expected to be announced Monday.
Observers happy
Observers from the African Union and the Arab League said they were happy with the way the vote was going, noting a "good atmosphere" and a "strong mobilisation of voters".
The proposed law, to replace a former constitution dating back to 1991, has already won the approval of most political players and local civic and political rights groups.
It would be the first in a series of major democratic reforms announced by the 17-member military junta council, which took power in a bloodless coup last
August ousting president Maaouiya Ould Taya, wants to introduce in a bid to end "more than 20 years of despotic rule."
Mr Ould Taya himself took office after a coup in December 1984 and was subsequently re-elected three times for six-year terms.
Under the proposed constitution, the maximum age for a president would be 75, the minimum 40.
It will also be mandatory for future heads of state to take an oath not to revise or back any efforts to change the law with respect to presidential terms.
The proposed constitution provides for a presidential regime giving the head of state major powers, including the appointment of the prime minister, though the parliament can vote no confidence in the government or censure it.
Among early voters was the leader of the junta, Ely Ould Mohamed Vali, who said he was confident of a huge vote in favour of the text.
A ‘great day’ for Mauritania
"This is a great day and a rebirth for Mauritania ... I am sure that all
Mauritanians feel the same way and that for this reason the constitution will receive massive support."
But he said: "Whatever the result I shall respect my promise" to hand back power to civilians in 2007.
Mauritanian authorities predicted a turnout of 80 percent, based on the takeup of voters' cards.
There have been fears that disillusionment after the long period in office of Mr Ould Taya and physical problems of getting to the polls might drive down the turnout.
Only two small parties, the Alliance for Justice and Democracy and the Party for the Third Generation, and an exiled group campaigning for the rights of black Mauritanians, have called for a boycott of the vote.
They argue that the new constitution ignores issues of "cohabitation between the different national communities", such as Arabs and black Africans, and does not address the thorny issue of slavery, which was officially abolished in 1981 but is still being practised in parts of the country, according to rights activists.
The referendum is to set into motion a long democratic process to be followed by municipal and legislative elections in November, as well as separate polls in 2007 to elect members of the upper house and a president.