Kate Leaves the Hospital After 4-Day Stay | Kate Middleton, Prince William

Prince William and Kate outside King Edward VII Hospital on Dec. 6

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Kate Leaves the Hospital After 4-Day Stay

Update Thursday December 06, 2012 11:00 AM EST Originally posted Thursday December 06, 2012 07:15 AM EST

With a smile and the briefest of comments that she is feeling "much better," a fragile-looking Duchess of Cambridge left the hospital Thursday morning.

An expectant Kate walked out of King Edward VII hospital with husband William at around 10:45 a.m.

Just before she stepped into her waiting car, Kate – in a navy checked overcoat and light blue scarf and clutching a bouquet of yellow roses – softly answered questions from the ranks of media crews about how she was doing.

She and William then headed to their cottage home in the grounds of Kensington Palace where she will continue to rest.

Kate, 30, had spent three nights and nearly four days at the clinic after suffering severe morning sickness in what is said by the palace to be the "very early stages" of her pregnancy.

In a short statement released by their spokesman as she left, the couple thanked the hospital for her care.

"The Duchess of Cambridge has been discharged from the King Edward VII Hospital and will now head to Kensington Palace for a period of rest," the statement read. "Their Royal Highnesses would like to thank the staff at the hospital for the care and treatment The Duchess has received."

William has been at the hospital every day throughout his wife's stay. On Wednesday, she had a visit from her sister Pippa, brother James and mother Carole.

"For the majority of women who have severe vomiting, things will get much better after about 12 weeks," Dr. Dagni Rajasingam, spokesperson for the Royal College of Obstetricians, tells PEOPLE. "The important thing is to keep hydrated – small quantities regularly throughout the day. If [an expectant mother] needs intravenous rehydration, they will require admission [to the hospital]. In this case, we hope it isn't as severe or prolonged as it could be."

Earlier this week, as news broke that Kate was in the hospital, a close friend told PEOPLE: "She's in the right place, in the right hands."

"God bless her," the friend said. "She is such a lovely person, and I hope everything is absolutely fine for her."

Reporting by SIMON PERRY