The 29-year-old yoga instructor checked into New York's Mount Sinai hospital on Friday where she gave birth to the couple's first child, who they named Carmen.
Hilaria tweeted: "We are overjoyed to announce the birth of our daughter Carmen Gabriela. She is absolutely perfect (sic)."
We are overjoyed to announce the birth of our daughter Carmen Gabriela. She is absolutely perfect. @ABFalecbaldwin @IrelandBBaldwin
— Hilaria Baldwin (@hilariabaldwin) August 23, 2013
55-year-old Alec's daughter Ireland, 17, from his marriage to Kim Basinger, also took to Twitter to share her joy.
She wrote: "Welcome to the world, Carmen! I'm a fucking sister!'' The best big sister!! Love you so much (sic)."
Welcome to the world, Carmen! I'm a fucking sister!
— ireland (@IrelandBBaldwin) August 23, 2013
Natural childbirth
A source told the New York Daily News: "Alec is overjoyed."
Hilaria previously revealed plans for a natural childbirth, although she admitted she could choose an epidural if the pain got too bad.
She said: "Right now, I want to do it naturally. That's not to say there's something wrong with the epidural, but that's the way I was raised - I'm a pretty natural person.
"But if it does get to the point where I am screaming for the epidural, I have no problems with that. We'll see where it goes."