The Duchess of Cambridge will “almost certainly” have her children at St Mary’s Hospital in Paddington, say bookies, helping firmly cement a new tradition for the royal family of having children in NHS hospitals.

St James’s Palace announced Kate’s pregnancy on Monday afternoon, as she was driven to the private King Edward VII’s Hospital Sister Agnes in Beaumont Street, Marylebone, from her parents’ Berkshire home.

She spent three nights in the hospital being treated for severe morning sickness, leaving with a smile yesterday and telling reporters she was feeling “much better”. But bookmaker William Hill said it was still overwhelmingly likely that Kate would choose to give birth in the private Lindo Wing of St Mary’s Hospital in Praed Street, Paddington, offering odds of 1-100.

The Lindo Wing, run by the NHS but open only to private patients, contains the neonatal unit of choice for both British and foreign royalty living in London.

Both Prince William and Prince Harry were born there, as were Zara and Peter Philips, the Princess Royal’s children. Princesses Beatrice and Eugenie, the daughters of the Duke and Duchess of York, were born in the private Portland Hospital in Great Portland Street, Regent’s Park, while older generations of the heir to the throne, including Prince Charles, have traditionally been born at Buckingham Palace.

Imperial College Healthcare NHS Trust, which runs St Mary’s Hospital and had its maternity services recently awarded a top safety rating, declined to comment.

But, barring an unexpectedly early birth, the decision will be one for William and Kate. A William Hill spokesman said: “The only reason why they might end up not going there is that they are in Anglesey or somewhere else away from London.”

William was at her bedside at King Edward VII’s Hospital Sister Agnes throughout her stay, and Kate’s sister Pippa also visited.

A St James’s Palace spokesman said the Duke and Duchess were “immensely grateful for the good wishes they have received”.

The couple were reportedly hoping to keep news of Kate’s pregnancy quiet until after Christmas, as she has not passed the 12-week point, but Monday’s announcement was prompted by the Duchess’s medical condition.

Kate is suffering a rare condition which can leave expectant mothers dehydrated and under weight. Known as hyperemesis gravidarum it can leave women unable to keep down food and fluids, and carrying twins may increase the chances of developing the illness.

The Prince of Wales has said he is “thrilled” that the Duchess was expecting, adding: “It’s a very nice thought to become a grandfather in my old age.”

She will now have a period of rest at her home at Kensington Palace, said St James’s Palace. But while there has been concern for the royal couple, there is excitement across the UK and beyond of news of a baby who will become third in line to the throne, regardless of its gender, with messages of support sent from leading figures both at home and abroad.

St James’s Palace would only say the couple became aware of the pregnancy “recently”. Her last public engagement on Friday, at her old prep school in Berkshire, saw the Duchess looking well, eating with pupils and staff and playing hockey in three-inch-high calf-length boots.

News that Kate’s symptoms may indicate she is pregnant with twins led William Hill to slash those odds from 33-1 to 7-1, while it also is offering even money that she will be “too posh to push” and give birth by C-section, or odds of 8-11that she is gives birth naturally.