"Mother and baby are fine" and the "Inception" actress is "very happy," a source told People.
Cotillard, 35, and Canet, 38, went public with the pregnancy news in January, but have said nothing about plans to marry.
The announcement of her pregnancy derailed the "Vie en Rose" Oscar winner's plans to appear in David Cronenberg's "Cosmopolis" with Robert Pattinson. The role was taken over by Samantha Morton.
Cotillard, whose "Midnight in Paris" premiered last week in Cannes and this week in New York and L.A., is, however, on board for the high-profile Batman movie "The Dark Knight Rises."
Canet was previously married to actress Diane Kruger.
-- Christie D'Zurilla
Photo: Marion Cotillard in September 2010. Credit: Darren Calabrese / The Canadian Press / Associated Press