Kate was taken by car to the private Lindo wing of St Mary's Hospital in Paddington, London, to give birth to her second child.
Loving God,as you watched over Mary at the birth of your beloved Son,graciously be with Catherine as she awaits the birth of her Royal Baby.
— John Sentamu (@JohnSentamu) May 2, 2015
The baby will be fourth in line to the throne after their brother, George.
Royal officials said her labour was progressing normally.
The Duchess of Cambridge was due at the end of April
Kensington Palace said: "Her Royal Highness The Duchess of Cambridge was admitted at 06.00hrs to St. Mary's Hospital, Paddington, London and is in the early stages of labour.
"The Duchess travelled by car from Kensington Palace to the Lindo Wing at St Mary's Hospital with The Duke of Cambridge."
Prime Minister David Cameron said: "My best wishes to the Duchess of Cambridge, who is having her second child today. The whole country will wish her well."
Christians have been tweeting well wishes and prayers on Twitter for the royals.
And so it begins.... prayers for a speedy and safe delivery for HRH http://t.co/Tm4pFPVCPM
— Sally Hitchiner ☩ (@SallyHitchiner) May 2, 2015