The couple's first child was born July 9.
"So happy! Just had a baby boy, Bingham 'Bing' Hawn Bellamy. Born 7lb 12oz, on 9th July," People.com quoted Bellamy as saying via Twitter. "Mum and baby are strong and healthy. Mum was a warrior, Bing popped out after 4.5 hours of intense pushing!"
Hudson also has a 7-year-old son from a previous marriage.
The "Nine" and "Almost Famous" star is the daughter of actress Goldie Hawn.