ByDaniela Relph, royal correspondent & Ali Abbas Ahmadi,BBC News
Reuters
The Prince and Princess of Wales are "enormously touched by the kind messages" they have received following Catherine's cancer diagnosis, a Kensington Palace spokesman has said.
Saturday's statement also added they were "grateful" that the public understood their request for privacy.
Catherine revealed in a video message on Friday that she had begun treatment.
She received her diagnosis after tests carried out following abdominal surgery "found cancer had been present".
A spokesperson for Kensington Palace said: "The Prince and Princess are both enormously touched by the kind messages from people here in the UK, across the Commonwealth and around the world in response to Her Royal Highness' message.
"They are extremely moved by the public's warmth and support and are grateful for the understanding of their request for privacy at this time."
The statement is the last we will hear from the prince and princess for a while.
It is a message that is both a thank you and a polite warning - with the couple saying the public side of this diagnosis is over and they want to be left alone.
It is also an acknowledgement of the overwhelmingly positive response to Catherine's video message.
Social media has been a place of such toxicity for the princess in recent weeks with the wild rumours and gossip about her condition.
At the time of writing, Catherine's video message, posted on Instagram, had been liked almost five million times.
Their latest "thank you" statement is the final chapter in this stage of the princess's diagnosis and Kensington Palace would like it to draw a line under all the speculation.
The past few weeks have been tough for the royal couple, their children and the team around them who have been trying to dampen down the gossip and rumour - a task that proved impossible.
The statement recognises the global interest in the princess's condition with a nod to the UK, the Commonwealth and the world.
But it is the last few words that are perhaps the most meaningful - they are a direct appeal for privacy as they try to escape the glare.
Prince George, Princess Charlotte and Prince Louis started their school Easter holiday on Friday.
They do not return to school until 17 April and we do not expect to see or hear from the family during the break.
They have made it very clear that this is family time that they will protect fiercely.
The central element of any royal Easter is the church service on Easter Sunday, which is usually a strong gathering of royals at St George's Chapel in Windsor.
Prince William and Catherine have already said they will not attend the service this year, but we may see King Charles III lead the family to church.
Buckingham Palace say they are "hopeful" he will be able to join the family next weekend.
The King has said he is "so proud" of his "beloved daughter-in-law" and that he and Camilla are in the "closest contact" with her.
And the Duke and duch*ess of Sussex are understood to have reached out privately to Catherine and William, also releasing a statement wishing her "health and healing" and the space to recover "privately and in peace".
Nearly all of Sunday's papers lead with reaction to news of the Princess of Wales' cancer diagnosis.
The Sunday Express wrote of "a nation touched" by the "courage and dignity" of Catherine while The Daily Star Sunday recalled some of the speculation about her health before the announcement, saying celebrities had now "rushed to apologise".
Royal commentator Paddy Harverson, who was previously the official spokesman of William and Catherine, told the BBC he did not think the institution of the Royal Family was more "fragile" because of the recent illnesses.
He told the BBC's Sunday With Laura Kuenssberg programme: "We just have to come to terms with the new reality, there's fewer of them.
"And so everyone just needs to sort of understand that they will still be busy, they will get over this. I am highly confident that the King, who I know well, is incredibly strong, very resilient, a great spiritual person, so I know he'll get through it.
"And likewise with the princess, with Kate, I think once they're through this sticky patch, I think we'll get back to normal."
You can watch a BBC News special programme about how the Princess of Wales revealed her cancer diagnosis in a video message to the nation - 'Kate's Cancer diagnosis' - on BBC iPlayer, now.
UK Royal Family
Prince William, Prince of Wales
Catherine, Princess of Wales
