The couple, who married last year in a private, low-key ceremony, have tried to keep their private life out of the media spotlight although Bardem paid tribute to their relationship when accepting an award at Cannes for his performance in Biutiful: "I share this joy with my friend, my companion, my love: Penelope: I owe you a lot and I love you so much."
Cruz gave birth to the child in Los Angeles' Cedars Sinai hospital but the couple are yet to announce the baby's name or gender.
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