Bruni revealed the newborn's name Thursday on her Web site.
"I am deeply touched by the many messages of congratulations that I have received since the birth of our daughter, Giulia," she said.
"On this happy occasion, my husband joins me in expressing our warmest thanks to all those of you who sent us these tokens of your kindness."
Sarkozy, the first French president to become a new father in office, was not on hand for the birth, Le Figaro reported. He spent about 30 minutes with Bruni earlier in the day and then rushed to a meeting in Germany with Chancellor Angela Merkel and other officials, returning to La Clinique de la Muette after the baby's arrival at about 8 p.m., Le Figaro said.
Sarkozy has two sons, both in their mid-20s, from his first marriage and another from his second. The baby is his first daughter and his first child with Bruni, whom he married in 2008 at the Elysee Palace.
Bruni, a model and singer, also has a child from a previous relationship.
In a television interview in September, Bruni said she planned to shield the new baby from publicity, People reported.
"I will do everything to protect this infant, and I'll be absolutely rigorous," Bruni told Radio France Internationale. "I will never show photos of this child. I will never expose this child."
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