It said there was a 99 per cent turnout for the vote, held in areas not affected by the cyclone, Al Jazeera online.
The announcement on Burmese state television came as the forecast for survivors of the cyclone is getting increasingly bleak.
Relief supplies still have not reached the people who need them most and now bad weather is threatening aid distribution, the online news agency said.
The military government's decision to press ahead with the May 10 poll, a week after the deadly cyclone hit, was sharply criticised by aid agencies who said the government should instead be concentrating its resources on helping cyclone survivors and preventing disease.
Regions devastated by the cyclone, are set to vote on May 24, even though the United Nations estimates that around two million people are still in desperate need of food, water and shelter.
Burma's government has said 66,000 are dead or missing from the cyclone, but the Red Cross has said it believes the death toll could be in excess of 100,000.
The new constitution, which took 14 years to draft, has been heavily backed by the military government and state media.
It says the 194-page document will form the basis for democratic elections to be held sometime in 2010.
Al Jazeera correspondent Tony Cheng, who crossed secretly into the Burma border town of Myawaddy during the Saturday's vote, found few people who had read the constitution or supported it.
Many people he spoke to said they planned to put an X in the 'No' box. Critics have denounced the constitution as a sham, designed only to institutionalise the military's grip on power.
Under its terms the military will be guarantied a quarter of all seats in a future parliament, while another clause allows the president to hand over all power to the military in a state of emergency.
Activists who have spent time in jail because of their opposition to the military government will be barred from standing for election because of their criminal records.