The Duchess of Cambridge gave birth to a baby son weighing 8lbs 6oz at 4.24pm at St Mary's Hospital in London.
Prince William is the president of the FA.
Greg Dyke, the FA chairman, said: "On behalf of The Football Association, I would like to congratulate their Royal Highnesses on the birth of their son. It is truly wonderful news.
"All of us at The FA wish the new family well and all the very best for the future."Prince William is an Aston Villa fan and the Barclays Premier League club have also offered their congratulations to the royal couple as well as sending a replica kit for the new arrival.
A statement on www.avfc.co.uk read: "Villa would like to extend congratulations to the Duke and Duchess of Cambridge on the arrival of their baby boy.
"The Duke is a huge Villa fan and it is hoped that his new son will be brought up in the claret and blue tradition.
"As a birthday present, Villa chiefs have sent the 8lbs 6oz tot a new kit with the appropriate letters 'HRH' and the number '1' on the back."