Marion Cotillard Gives Birth to a Son
05/20/2011 AT 07:00 AM EDT
Their son, named Marcel, was born Thursday night in Paris, a source confirms to PEOPLE. "Mother and baby are fine," says the source, who adds that Cotillard is "very happy."
Cotillard, 35, and Canet, 38, first starred together in 2003's Love Me If You Dare and began dating in 2007.
Though Cotillard – who won a Best Actress Oscar as singer Edith Piaf in 2007's La Vie En Rose, and will costar with Anne Hathaway and Christian Bale in the upcoming Batman movie, The Dark Knight Rises – has been spotted wearing a diamond solitaire from Canet on her left hand, they are not engaged.
"They haven't disclosed any plans concerning marriage," a source told PEOPLE when the pregnancy was announced in January.
The couple most recently collaborated for the film, Little White Lies, which Canet directed.
Cotillard currently costars in Woody Allen’s Midnight in Paris, which kicked off this year's Cannes Film Festival on May 11 – Cotillard did not attend – and is opening in limited release in Los Angeles and New York on Friday.