The Portugal international, 37, and Rodriguez, 28, were expecting twins.
Their baby girl survived, and they said her birth "gives us the strength to live this moment with some hope and happiness".
"Our baby boy, you are our angel. We will always love you," they wrote.
In a statement on social media, they said: "It is with our deepest sadness we have to announce that our baby boy has passed away.
"We are all devastated at this loss and and we kindly ask for privacy at this very difficult time."
The pair had announced the pregnancy in October.
