Prince William is all smiles - but Kate could be on a drip for monthsPA

Prince William was all smiles last night as he emerged from hospital after seeing his pregnant wife.

The Duchess of Cambridge has been left exhausted by the acute morning sickness, or hyperemesis gravidarum, that is blighting her pregnancy – and could face weeks on a drip – but her condition has clearly improved enough to give her proud husband something to smile about.

In a statement on Tuesday evening, a St James's Palace spokesman said: "The Duchess of Cambridge is continuing to feel better. She and the Duke are immensely grateful for the good wishes they have received. She will remain in hospital at present."

The Duchess, who was admitted to the hospital on Monday, is thought to be less than 12 weeks pregnant.

Although extreme morning sickness does not affect the health of the baby, it could last the whole of Kate's pregnancy.

Kate, 30, has been on a drip aimed at rehydrating her since she was admitted to the King Edward VII hospital in Central London on Monday.

But it has been revealed that while most of the nation has been wishing her well, some people have been using the Duchess's condition to play practical jokes after it emerged that hoax callers from a radio station in Australia got information from the hospital treating Kate.

King Edward VII Hospital in London received a call from two presenters from Sydney radio station 2Day FM.

Presenters Mel Greig and Michael Christian posed as members of the Royal family - believed to be the Queen and Prince Charles - and were transferred to a nurse who gave them an update on Kate's condition. It was played out live on the radio show.