The figure, higher than an initial three-week estimate of 10,400 deaths from the country's leading undertaker, was compiled from death certificates, health officials said.
But top medical officer Gilles Brucker said the final toll could be much higher.
"These figures do not show the full extent of the problem," he told France 2 television, adding that the total for all of August should be available in late September.
The official figures released on Friday exceed any heatwave deaths recorded in France's European neighbours and showed around 55% more deaths than would have been expected.
Theories as to why range from cuts in services for the elderly to many old people being left alone while offspring went on holiday, as many French people do in August.
The issue has even forced the cash-strapped government to consider axing a public holiday to fund better care for the elderly.
Pensioners were hit hardest by the worst heatwave since records began in 1945, as temperatures soared above 40 degrees Celsius, causing dehydration and hypothermia.
French health minister Mattei
(2nd R) has resisted calls to resign
"The human drama of the heatwave has hit the most fragile in our society...I am determined more than ever to make changes to our health and welfare system," said Health Minister Jean-Francois Mattei, who has resisted calls to resign.
The soaring summer toll shocked France, whose health service came out on top in a World Health Organisation survey in 2000.
On Friday, Le Parisien daily published a poll showing 70% of voters backed a plan to cancel one public holiday to finance a special fund for pensioners through increased tax revenues that would result from greater output.
The Pentecost Monday holiday, a moveable feast which usually falls in May, is seen as most likely for the chop, a move that has already received the blessing of the Catholic Church.
The government was sharply criticised for failing to react fast enough to the crisis which swamped hospitals and morgues.
There are still many bodies which remain unclaimed in refrigerator trucks and even a cool hall in a wholesale food market outside Paris – which were commandeered as temporary morgues.
Scores of bodies have been buried in temporary graves by the authorities as a measure until relatives of the dead come forward.
French media reported that Paris City Hall at one stage reckoned the capital had around 200-300 unclaimed bodies.