"It's a girl. The tiny new empress of my everything . She's immaculate. We're delirious," he writes.
The couple — who announced they were expecting in May — were married in 2004, but it was only recently that the Perception star, 33, and Gillies felt ready to bring a baby into the mix.
"We're both people who like kids, but we weren't jumping up and down, like, 'Oh my God, let's start that chapter of our lives,'" Cook told PEOPLE in June. "There was just so much more both of us wanted to do."