The couple plans to call their daughter Suri, according to a statement from publicist Arnold Robinson, who insists "mother and baby are doing well."
Cruise recently revealed he and his fiancee arrived at a name for their child after coming across it in a book.
He said, "Friends of ours gave us a book and we just opened it and instantly came up with the name."
Ironically, baby Suri was born on the same day as Brooke Shields' new daughter Grier. Cruise and Shields launched a war of words last year when the Scientologist actor criticised the Blue Lagoon star for publicly championing medication that helped her overcome post-natal depression following the birth of her first child.
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