The `Homeland` actress and her British actor husband Hugh Dancy welcomed their first child Cyrus Michael Christopher Dancy into the world on 17.12.12.
A spokesperson for the Emmy award winning star confirmed news of the baby`s arrival to People magazine.
The 33-year-old blonde beauty confirmed her pregnancy in July and revealed she was determined to keep the baby`s sex a secret.
She also said she was looking forward to taking a break over the festive season and "getting to know our new Dancy".
Claire married Hugh, 37, in a private ceremony in France in 2009 and recently revealed she confides everything in the actor, even though they were forced to spend a lot of time apart recently because of their respective working commitments.
She said: "I haven`t been spending a lot of time with Hugh lately because we`ve been in so many different places with work, but he is the person I tell absolutely everything to. I trust him more than everyone else.
"A couple of weeks ago I went on a date night with Hugh, who I hadn`t seen in a long time because of work. "We went out to a restaurant in New York and I had a big bowl of pasta. I was a little overwhelmed with joy to be with my guy, eating lots of carbs. It was heaven."