Prince William said the couple "could not be happier" following the birth of the boy, who weighed 8lb 6oz and is yet to be named, at 16:24 BST on Monday.
Thousands of well-wishers descended on Buckingham Palace after the news broke.
The royal birth will be marked later with gun salutes and the ringing of Westminster Abbey's bells.
The duke was at the private Lindo Wing of St Mary's Hospital, west London, for the birth - and stayed with Catherine and the baby overnight.
A Kensington Palace spokesman said it was "too early" to say whether they would leave hospital on Tuesday. The couple are expected to talk to their medical team before a decision is made.
BBC royal correspondent Peter Hunt said it was unlikely the Queen would visit her great-grandchild in hospital, adding - "she can of course see him privately once he leaves". (+ video) » | Tuesday, July 23, 2013
SPIEGEL ONLINE: Royaler Nachwuchs: Baby Cambridge, neuer Werbeträger der Monarchie – Erst die Hochzeit, nun das Baby: Prinz William und Kate sind die Hoffnungsträger der angestaubten britischen Monarchie. Die jungen Vorzeigeeltern machen die Firma Windsor wieder zukunftsfähig. » | Von Carsten Volkery | Montag, 22. Juli 2013
LE POINT: GRANDE-BRETAGNE - Le "royal baby" sera 3e dans l'ordre de succession au trône : Kate, duchesse de Cambridge et épouse du prince William, a accouché lundi d'un garçon. Les messages de félicitation affluent du monde entier. » | Le Point.fr | mardi 23 juillet 2013
