TRH Duke & Duchess of Cambridge
London SW1A 1BA
Why Clarence House, not Kensington Palace?
For more info on Writing to the Royals check out our helpful guide. It has everything you need to know.
They should reply with their usual postcard sized photo with typed message on the back. Expect longer response times, as they will be getting a lot more mail than usual! (With Charlotte I know it took them at least 4 months to send out all the replies, and could very well be longer with all the mail for Harry's wedding!)
Their response from Charlotte's birth: