Voting officials estimated the turnout was high in a referendum across the vast central African country, which has been emerging since 2003 from its last devastating conflict, and is due to hold elections by the end of next of June.
The Independent Electoral Commission extended voting for a second day into Monday because of long queues at some of the 35,000 polling stations.
They reported few irregularities, as did some 5,000 domestic and 280 foreign observers.
No results were available late Monday due to the lack of “significant statistics”, electoral commission head Apollinaire Malu Malu told reporters.
“The integrity of the electoral process requires us to disappoint you this evening. It would be hazardous to put forward figures before the arrival of significant statistics,” he said without revealing when the first results would be announced.
The constitution paves the way for elections for a president, who will have a once-renewable five-year term, and a bicameral parliament, also for five years.
Most observers expect a “yes” vote at the national level, which has been encouraged by President Joseph Kabila, former rebels who fought his army in 1998-2003 but are now part of an interim government, and most politicians.
But the capital of Kinshasa, where the opposition is strongest, could reject the charter.
United Nations-supported Radio Okapi said results posted during counting at polling booths suggested the “yes” vote nationwide was outstripping the “no” by 72 percent to 16 percent.
“A ‘no’ in Kinshasa would be sad because it would give force to the divide between the capital, where people tend to overestimate their political clout, and the rest of the country, particularly in the east, which bore the brunt of the war and wants to be done with the transition at any price,” one local observer said.
The vote was the first of real significance since independence in 1960 for a nation rich in precious minerals and other natural resources which has endured a long, thieving dictatorship, two recent wars embroiling regional armies, and considerable foreign interference.
On Sunday, turnout was highest in eight provinces, according to the electoral commission, which declared itself “satisfied overall” with the vote on a text that would change the political system and also divide the country into smaller administrative districts.
In Kasai Occidental and Kasia Oriental, two central provinces where there is support for veteran opposition politician Etienne Tshisekedi, people appeared initially to have followed his boycott call, observers indicated.
Turnout was also low in Kinshasa, but people who said they had hesitated before rushed to cast their votes on Monday.
Without giving figures, the electoral commission put rough overall turnout at more than 60 percent.
The final outcome, in a country that stretches more than 1,300 kilometres from the capital in the west to the volatile eastern borders, is not expected for several days.