Fox India Owen was born on Wednesday, weighing 7lb 1oz.
He and wife Emma already have a son, Elwood, and a daughter, Willow.
Mark's bandmate, Gary Barlow, is due to become a father for the fourth time in the coming weeks.
A statement issued on behalf of Mark said: "Mark, Emma, Elwood and Willow are delighted to announce the arrival to the clan of their beautiful new baby girl Fox India Owen.
"They would like to thank all their friends and family for all the wonderful messages."
Fellow Take That star Robbie Williams is also imminently to become a father. He and wife Ayda are expecting their first child.
Gary was among those to send his good wishes to Mark today. In a message on Twitter, he said: "Massive congratulations to the Owens on the birth of their daughter Fox."