- The princess said it had been an "incredibly tough couple of months" for her family but she was "well and getting stronger every day".
- "His Majesty is 'so proud of Catherine for her courage in speaking as she did'," Buckingham Palace said.
King Charles III is "so proud of Catherine for her courage" in speaking about her treatment for cancer, Buckingham Palace has said.
The King, who has been diagnosed with cancer himself, is in the "closest contact with his beloved daughter-in-law", a short statement read.
Support has poured in for the princess, who described her diagnosis as a "huge shock" in a video released on Friday.
She has been undergoing preventative chemotherapy since late February.
The princess said it had been an "incredibly tough couple of months" for her family but she was "well and getting stronger every day".
Her news comes six weeks after King Charles paused his public appearances following a cancer diagnosis himself.
The two were briefly treated at the London Clinic private hospital at the same time.
Catherine had a planned abdominal surgery there in January, and the King was admitted for a "corrective procedure" for an enlarged prostate.
"His Majesty is 'so proud of Catherine for her courage in speaking as she did'," Buckingham Palace said.
King Charles and Queen Camilla, who were informed of the news about the princess's health before Friday's announcement, said they would "continue to offer their love and support to the whole family through this difficult time".
The Duke and Duchess of Sussex sent a message saying: "We wish health and healing for Kate and the family, and hope they are able to do so privately and in peace."
Catherine's brother James Middleton posted a message of support for his sister on Instagram, writing: "Over the years, we have climbed many mountains together. As a family, we will climb this one with you too".
Kensington Palace has not disclosed the type of cancer Catherine has been diagnosed with, but says it is confident she will make a full recovery.
Members of the public have been sharing messages of sympathy with the princess.
Eve Watts, who lives in Windsor, said the news was a "shock because she's so young". She added: "It just shows you it could happen to anyone."
Ms Watts' mother Valerie, who was visiting the historic town from Southport, said she hoped Catherine's video would "end all the speculation that's been going on".
Across the road from Windsor Castle, visitor Sonia Dunn said the princess was an "amazing young woman" and a "huge asset to the country", adding: "We just need to let her have time and space."
Prime Minister Rishi Sunak said on Friday that Catherine had shown "tremendous bravery" in the face of "intense scrutiny".
He added: "When it comes to matters of health, like everyone else, she must be afforded the privacy to focus on her treatment and be with her loving family."
Labour leader Sir Keir Starmer said his thoughts were with the Royal Family, and that he was heartened by the princess's optimism and "message of faith and hope".
Sir Keir also referenced the scrutiny Kate had faced in recent weeks, remarking on the "added stress" of receiving her diagnosis amid "lurid speculation".
Meanwhile, US President Joe Biden said he was joining "millions around the world in praying" for Catherine's "full recovery".
French President Emmanuel Macron said the princess's "strength and resilience inspire us all".
Catherine's video statement explains that when she had abdominal surgery earlier this year, it was not known that there was any cancer.
In January, Kensington Palace announced that the princess had undergone a planned surgery which was not cancer-related and that she would remain in hospital for 10-14 days before continuing her recovery at home.
On Friday, the princess said tests following that operation "found cancer had been present".
"My medical team therefore advised that I should undergo a course of preventative chemotherapy and I am now in the early stages of that treatment," she added.
The 42-year-old said she was thinking of all those who have been affected by cancer, adding: "For everyone facing this disease, in whatever form, please do not lose faith or hope. You are not alone."
Catherine and the Prince of Wales are not expected to appear with the Royal Family on Easter Sunday.
There will not be any early return to official duties for the princess either.
Kensington Palace has also said Prince William's sudden absence from a memorial service on 27 February was because of the discovery of Catherine's cancer diagnosis.
The couple have faced intense public speculation and a social media frenzy about her health - including conspiracy theories - since her operation in January.
She has not attended any official events since Christmas.
Earlier in March, the prince and Catherine released a photo of the princess with their three children to mark Mother's Day. The image was later withdrawn by several news agencies over concerns it had been "manipulated".
Kensington Palace later released a statement from Catherine apologising "for any confusion" over the image, and explaining that she occasionally experiments with photo editing.
While Catherine has stood back from royal duties, Charles has continued with his constitutional role as head of state, and was recently seen in a virtual address to a service celebrating Commonwealth Day.