The Duke and Duchess of Cambridge have left Kensington Palace with their four-day-old daughter Princess Charlotte and her elder brother George for their Norfolk home Anmer Hall.
The Duke and Duchess of Cambridge named their daughter Charlotte Elizabeth Diana in honour of the Queen and also Prince William's mother.
David Cameron said he has his "fingers crossed" for a healthy and happy baby as the Duchess of Cambridge is in the early stages of labour.
Asked about the expectant father Prince William, Cameron said: "I think he's a pretty relaxed dad and he seems to take everything in his stride.
"He's a remarkable young man, they are an amazing couple and we all just wish them well," the Prime Minister said.
The Takasakiyama Natural Zoological Garden in Oita is now considering renaming Charlotte after complaints from the Japanese public.
Kensington Palace warning over 'harassment' around Sandringham where Kate and William are staying with princess Charlotte and prince George.
Naming the royal baby Charlotte Elizabeth Diana will be seen as a tribute to the Duke of Cambridge's father, mother and grandfather.