The healthy baby girl was born on Friday night – and the couple have named her Delta.
Proud dad Dax, 39, announced her arrival on Twitter, joking: "Delta Bell Shepard is here. She doesn't look like either Kristen or me, but we're gonna keep her anyway."
Kristen, 34, who voiced the adult Princess Anna in the blockbuster movie, wrote on Twitter: "The estrogen has now hit critical mass! Good luck, Dax. And welcome to the world DELTA BELL SHEPARD."
She accompanied the Tweet with a photo of an ape's hand tenderly clutching a baby ape's.
Earlier, she had written: "Crying in the car to @joshgroban christmas songs has got to be an early sign of labor, right?"
Delta is a second child for the couple, joining 21-month-old big sister Lincoln.
Kristen and Dax had earlier talked of their delight at finding the 'perfect' name for their second child.
According to the Parentdish Baby Names Generator, Delta means 'mouth of a river', and is a letter of the Greek alphabet. The most famous owner of the unusual moniker is Australian soap star Delta Goodrem. Delta is going to receive a warm reception from her big sister.
Singer Kristen, who also starred in Parenthood, said recently: "She's very interested in the shape of my belly. Very interested in offering it water and crackers and giving it smooches.
"She could just be mimicking what we're doing but I have a hunch she knows what's happening."
You can hear Kristen in Frozen on Sky Premier on Christmas Day.