"It happened between 7:00 and 7:30 pm. Everything went well," a second source told AFP. The staff at the clinic has been reminded of the strict duty of medical confidentiality, and Sarkozy's office was silent.
The girl is the French first couple's first baby since their marriage in 2008, although Bruni has one 10-year-old son from a previous relationship and thrice-married Sarkozy has three aged 14 to 26.
The baby is also the first born to a serving French president.
In an interview pre-recorded for a French state television broadcast due on Thursday, Bruni-Sarkozy said she had not known the baby's sex: "We've arranged for it to be a big surprise. Obviously, a nice surprise."
Sarkozy had earlier in the day visited his 43-year-old wife in the Paris clinic for half-an-hour, but was forced to jet off to Germany for crisis talks on the eurozone sovereign debt crisis before the birth itself.
He left Frankfurt just before the birth was announced, after talks with Chancellor Angela Merkel, IMF director Christine Lagarde and the current and next heads of the European Central Bank Jean-Claude Trichet and Mario Draghi.
Sarkozy's Elysee Palace office has said it will not make any official announcement on what it considers a private family matter, but the birth of a daughter was widely reported in French media outlets.
The French president left the hospital shortly after midnight without talking to the assembled reporters.
The unpopular 56-year-old centre-right leader is languishing in the opinion polls, far behind newly-crowned Socialist challenger Francois Hollande, and faces a tough battle for re-election in next April's first round vote.
But -- despite the lure of a rare round of warm-hearted headlines -- the Elysee has so far chosen not to publicise the pregnancy, fearing that France's electorate would react badly to the exploitation of private life.
In any case, political scientists told AFP that any "baby bump" in the opinion polls would be short-lived and arrive too late to in itself save Sarkozy from the ignominy of a single-term presidency.
This was perhaps why he chose to miss the opportunity of seeing his first daughter's birth in order to head to Frankfurt and battle for what could be a greater political prize, that of saving the European financial system.
In the run-up to the birth, Bruni-Sarkozy insisted political considerations and timing -- even the expectation of a birth in mid financial crisis -- had not been taken into account as the first family made its plans.
She insisted any baby's arrival is "a happy carefree moment and that's how it's been since the dawn of time.
"We're in a time of crisis, but if human reproduction was decided by thinking about whether you're going to have a perfect life, we wouldn't be here to talk about it, neither you nor I," the Italian-born heiress said.
"What's more, I think that the survival instinct is also expressed by the desire to have a child."
"Of course, I'll look after the baby, but I don't see that that should stop me working," she said, stressing that she receives a lot of help and does not have a difficult life.
"As to the duties related to my husband's job -- and there are not so many -- I do them willingly," she said.
Bruni-Sarkozy dodged a question about whether Sarkozy would take part in next year's presidential campaign, which the journalist suggested might be "ferocious", saying: "Me, I don't campaign!"
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