The 33-year-old gave birth to a son, Cyrus Michael Christopher Dancy, on Monday.
It's the first child for the 'Homeland' star and her British actor hubby, Dancy who married in 2009.
Hugh and a pregnant Claire at this year's Emmys
It tops off a huge year for the actress who was nominated for a Screen Actors Guild Award and a Golden Globe for her role in 'Homeland'.
She's up for the Best Performance by an Actress in a Television Series gong at the Globes ceremony on 13 January.
Who's the mumma? Peaches Geldof Who's the daddy? Thomas Cohen Baby ETA: April 2013 Interesting fact: Peaches and Thomas are expecting a boy and are planning to name him Phaedra.
Who's the mumma? Fearne Cotton Who's the daddy? Jesse Wood Baby ETA: Early 2013 Interesting fact: Fearne had to deny she knew the sex of her baby when she referred to her bump as 'her'.
Who's the mumma? Imogen Thomas Who's the daddy? Adam Horsley Baby ETA: February 2013 Interesting fact: Imogen and her boyf had been dating 10 months when the star fell pregnant