According to an account in the Spanish newspaper El Pais, the child was born at Cedars Sinai Hospital in Los Angeles. The name and gender of the child have yet to be made public.
Cruz first publicized her pregnancy last September while filming the latest edition of the Pirates of the Caribbean franchise.
Top Entertainment Photos
Bardem and Cruz were secretly wed in July of 2010 in the Bahamas in front of a small group of family and friends.
Bardem was just nominated for a Best Actor Oscar for his role in Biutiful. He previously won a Best Supporting Actor award for playing killer Anton Chigurh in No Country for Old Men. Cruz also has her own Oscar for Best Supporting Actress for her work in Vicky Cristina Barcelona.