Firefighters said that 28 other people, including a firefighter, were slightly injured in the midnight blaze in southeast Paris.
Local officials had said earlier that up to half the dead were children. Many of the victims were from the west African nation of Mali. Others were from Senegal, Ghana and Tunisia, according to building residents.
"It's an extremely heavy toll," said Interior Minister Nicolas Sarkozy who visited the scene of the blaze in southeast Paris.
The children had been "asphyxiated," he said. "It's an abominable spectacle."
The minister blamed the high death toll on overcrowded conditions.
The district mayor, Serge Blisko, said the dead had "visibly, died in their sleep, asphyxiated and not burned."
One resident of the seven-story building, situated on the corner of a major boulevard, described being awakened by cries from children and adults, then rushing to his window on the building's second floor.
People "jumped out the windows. They didn't care about dying," said 71-year-old Oumar Cisse, originally from Mali.
"This dreadful catastrophe plunges all of France into mourning," said a statement from President Jacques Chirac. He asked that the cause of the blaze be determined as quickly as possible so that "all the consequences can be drawn."
The fire broke out shortly after midnight in the stairwell of the building, said Capt. Jacques Dauvergne, spokesman for the firefighters. About 210 firefighters worked for an hour and a half to bring the blaze under control. It was extinguished after burning through the upper floors of the building for some three hours, Dauvergne said.
Police were investigating the cause, taking samples from the building for laboratory analysis, Dauvergne said.
Portions of the facade of the large white building on a corner of a major boulevard in the city's 13th district were scorched from the smoke, but the building was intact. Window boxes with flowers abloom that fell from balconies were strewn on the sidewalk below.
About 100 children and 30 adults lived in the building, which was run by a humanitarian association that helps needy people, according to Cisse who has lived in the building for 15 years.
Cisse said the building was in a decrepit state, infested with rats and mice.
Walls were cracked and lead was in the paint that covered them, he claimed.
Electricity bills were "exorbitant" for some residents, he added, suggesting that the wiring could be faulty.
"It was totally unfit," said Cisse, who acts as a go-between for residents and an association that manages the building.
Blisko, the district mayor, said the building was "overcrowded," particularly with children.
"They talk about three-room apartments with 12 people," Blisko said in an interview. "When you have this type of fire and people are sleeping, you can be sure the toll will be high."
Sory Cassama, who lived in the building with his wife and 12 children, said he was asleep when a daughter knocked on the door. Their living room had filled with smoke.
"There was so much smoke in the stairwell, but we were still able to get out," said Cassama, who said his wife was hospitalized with smoke inhalation.
The state-owned building was run by the humanitarian organisation Emmaus and was under the direct care of a linked association France-Euro Habitat, according to local officials and building residents.
"We have enormous difficulty housing the disadvantaged," Blisko, the mayor, said.
Sarkozy, the interior minister, said he has asked that all such buildings be inventoried with an eye to closing some.
It was the second major Paris fire this year.
In April, 24 people were killed in a hotel blaze in the French capital - many of them children. Most were African immigrants and other people without means who were lodged there by authorities.
In that fire, officials said a night watchman's girlfriend may have accidentally triggered the blaze by placing candles on the floor to set the scene for a romantic tryst but then leaving in a rage because he was drunk.