A spokeswoman for the actress confirmed Wednesday that Cruz gave birth to a boy last week. She says "both parents and baby are doing great." She provided no other details.
The couple were married at the beginning of July in the Bahamas.
The 36-year-old Cruz and the 41-year-old Bardem are from Spain.
They appeared together in the 2008 Woody Allen romantic comedy "Vicky Cristina Barcelona." Cruz won a supporting-actress Oscar for her role.
Bardem is nominated for best actor this year for his role in the Spanish-language drama "Biutiful." He won a supporting-actor Oscar for the 2007 crime thriller "No Country for Old Men."