The name was announced by Carla Bruni-Sarkozy on her official webstie on Thursday afternoon and confirmed earlier media speculation over the choice.
According to the French daily Le Figaro, the Sarkozys favoured the name as it can work in French, Italian and Hungarian - representing the new-born child's familial roots.
France's President Nicolas Sarkozy on Thursday described his "profound but private joy" at the birth of his daughter after visiting his wife Carla Bruni-Sarkozy and the newborn at a Paris clinic.
The baby girl was born late on Wednesday, and the president -- who missed the birth to attend talks in Germany on the eurozone crisis -- dropped in before midnight and then again for 50 minutes on Thursday morning.
"We have been lucky enough to find great happiness. All the parents here understand our very profound joy, a joy all the deeper because it is private," Sarkozy said later on a factory visit in the western Mayenne region.
The first couple had decided not to officially announce the birth, but the president reacted warmly when waste-processing plant workers gave him gifts.
"Oh, that's really kind, very thoughtful," he smiled, telling workers who asked after his wife and child: "They're doing really well."
Asked what the child would be called, Sarkozy said: "We're going to let Mum have the pleasure of revealing that."
Sarkozy continues to have a full political schedule, battling to rescue the economy and his own hopes of re-election.
With seven months to go, Thursday's latest BVA opinion poll predicted his political career will be buried in a Socialist landslide, with opponent Francois Hollande beating him by 64 to 36 percent in a second-round run-off.
Asked whether the birth would influence the election, with perhaps a "baby bump" in Sarkozy's polling numbers, Defence Minister Gerard Longuet said: "In any event, it will have an impact on Nicolas Sarkozy. He's very happy".
"A president comfortable in his skin, clear in his head and in his life, is a gift for the country," he added, speaking to the i-Tele news network.
Bruni, a 43-year-old Italian-born heiress turned supermodel turned singer, was famous even before she married Sarkozy in early 2008 after a whirlwind romance, and the birth made headlines around the world.
Russia's President Dmitry Medvedev and Prime Minister Vladimir Putin warmly congratulated Sarkozy. "I wish with all my heart for your baby girl to grow up healthy and happy and to please her parents with her success," Putin wrote.
But in France itself, where the public is accustomed to politicians' private lives staying private, the event was accorded a more muted reception, with some newspapers barely covering it at all.
In less than five years, Sarkozy has become the first president of France's modern Fifth Republic to divorce while in the Elysee, the first to remarry and now the first to have a child.
In 2007 when his relationship with his second wife Cecilia Ciganer-Albeniz broke down and he quickly took up with the glamorous Carla, voters were turned off by the very public display of his romantic adventures.
The lovers allowed themselves to be photographed with their children on luxury foreign holidays, reinforcing Sarkozy's growing reputation as a showy friend of the super-rich, out of touch with French life.
Sarkozy seems to have learned the lesson of that period, and has been extremely discreet about the pregnancy. Both the president and first lady have said they will not parade the child for photographers
The left-wing Liberation newspaper only mentioned the birth in a brief page 14 story. Others gave it more prominence but it was far from a national event, drowned out by financial crisis and the death of a French hostage in Somalia.
The Elysee never announced the pregnancy officially, but carefully outed Bruni as pregnant by allowing her to appear with a visible bump at an event to welcome fellow first ladies to the G8 summit in Deauville in May.
She has since given a handful of interviews, but always been careful to insist that her family life with Sarkozy has no bearing on his presidency.
Bruni has a 10-year-old son from a previous relationship and thrice-married Sarkozy has three sons aged between 14 and 26.
Bruni has said she will continue working on her singing career after the birth, and Sarkozy seems unlikely to take France's two-week paternity leave.
He left Bruni's side just before the birth to go to Frankfurt to seek a deal with Germany's Chancellor Angela Merkel to reinforce a eurozone bail-out fund and protect French banks from an eventual Greek default.
He arrived for his visit to the clinic on Thursday clutching a sheaf of paperwork, and immediately afterwards resumed his campaign schedule.