Ronaldo revealed in a social media post last October that the couple were expecting twins.
In a post released on the Manchester United forward's Twitter account, they confirmed the birth of a baby girl.
"It is with our deepest sadness we have to announce that our baby boy has passed away. It is the greatest pain that any parents can feel," Ronaldo and Rodriguez said in a jointly-signed statement.
"Only the birth of our baby girl gives us the strength to live this moment with some hope and happiness. We would like to thank the doctors and nurses for all their expert care and support.
"We are all devastated at this loss and we kindly ask for privacy at this very difficult time."
The couple, who met during Ronaldo's time at Real Madrid, have a four-year old daughter together, while Ronaldo has three other children.
"Your pain is our pain, Cristiano," Manchester United tweeted. "Sending love and strength to you and the family at this time."