She is believed to be attached to a drip providing food and fluid to combat the effects of dehydration caused by the condition.
Prince William's happy face when he emerged from London's Edward VII hospital after six hours by Kate's bedside showed the world she was feeling better.
She is believed to be about eight weeks pregnant although St James's Palace is still refusing to confirm this.
A Palace statement said the Duchess was feeling better and she and the Duke are very grateful for the support they have received.
The good news comes as it's being reported that Kate's sister Pippa (29) will be godmother to the baby.
Prince William's brother Harry, who is currently on tour in Afghanistan, is delighted with the news and is looking forward to becoming an uncle.
Meanwhile a new law is to be rushed through Parliament to ensure that if Kate has a daughter, the child will be an heir to the throne.
Commonwealth leaders finally reached agreement yesterday to end the centuries old discrimination against women in the British monarchy's line of succession.
The couple's first baby will be third in line to the throne, dropping Prince Harry down to fourth place -- or fifth if there are twins.
The vomiting condition which led to Kate's hospitalisation, could continue for the first 16 weeks of her pregnancy and she may need to take anti-sickness medication until it clears.
The Harley Street doctor who will oversee the pregnancy, Alan Farthing (47), is the former fiance of murdered TV star Jill Dando.
As the pregnancy progresses William and Kate plan to hide away from the limelight in the new year to make preparations for the new arrival.
There is a growing expectation in royal circles that Prince William will step down as an RAF search and rescue helicopter pilot in the middle of next year.