Carla also took the opportunity to reveal the little girl's name: Giulia.
The French first lady wrote:
'I am deeply touched by the many messages of congratulations that I have received since the birth of our daughter Giulia.
'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."
Mr Sarkozy had spoken earlier in the day of his "very profound joy" at the birth of his daughter.
Although the couple had chosen not to formally or officially announce Giulia's arrival, Mr Sarkozy spoke about his new daughter as he toured a waste-processing plant in Mayenne. Workers there gave him gifts for Giulia, and he told them his wife and baby were 'doing really well'.
He added:"We are lucky ... All parents here can understand our very profound joy, a joy all the deeper because it is private".
Carla has already said she is determined to keep the new baby away from the media spotlight - do you think she will succeed?
Check out the famous parents who've had babies this year:
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