The 39-year-old, from Ammanford in Carmarthenshire, revealed the news live on BBC One's The One Show on Thursday evening but has yet to reveal the name.
It is Miss Jones' first child with partner Charlie Thomson and he was born in the morning of Sunday, 22 January, weighing 7lbs 11.
Ms Jones was concerned she had left it too late to have a baby after filming a documentary, Fertility and Me.
Welsh BBC TV presenter Alex Jones has given birth to a baby boy.