By Sean Coughlan, Royal correspondent
Photo: AFP/Chris Jackson
The Princess of Wales says she is in the early stages of treatment after cancer was found in tests.
In a video statement, Catherine said it was a "huge shock" after an "incredibly tough couple of months".
But she sent a positive message, saying: "I am well and getting stronger every day."
Details of the cancer have not been disclosed, but Kensington Palace said it was confident the princess would make a full recovery.
The princess's statement explained that when she had abdominal surgery in January, it was not known that there was any cancer.
"However tests after the 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," the princess said.
The chemotherapy treatment began in late February. The palace said it would not be sharing any further private medical information, including the type of cancer.
The princess, 42, said she was thinking of all those who have been affected by cancer.
"For everyone facing this disease, in whatever form, please do not lose faith or hope. You are not alone."
Catherine said recovery from her surgery in January, for a condition which has not been revealed, had taken time and the priority was now reassuring her family.
"William and I have been doing everything we can to process and manage this privately for the sake of our young family.
"It has taken us time to explain everything to George, Charlotte and Louis in a way that is appropriate for them, and to reassure them that I am going to be okay."
She said the family now needed "some time, space and privacy".
King and Queen informed
The King and Queen had been informed of the news about the princess's health prior to Friday's announcement - and King Charles himself has also been undergoing treatment for cancer.
King Charles and Catherine were briefly treated at the London Clinic private hospital at the same time - Catherine had her abdominal surgery there and the King was admitted for a "corrective procedure" for an enlarged prostate.
A Buckingham Palace spokesperson said the King was "so proud of Catherine for her courage in speaking as she did".
After their time in hospital together, he has "remained in the closest contact with his beloved daughter-in-law throughout the past weeks".
Royal couple not expected at Easter gathering
Catherine and Prince William are now not expected to appear with the royal family on Easter Sunday, and there will not be any early return to official duties for the princess.
Kensington Palace 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, since her operation in January. She has not attended any official events since Christmas.
In her video statement, she spoke about the support from her family: "Having William by my side is a great source of comfort and reassurance too.
"As is the love, support and kindness that has been shown by so many of you. It means so much to us both."
Kensington Palace said the video of the princess was filmed on Wednesday by BBC Studios, the production arm of the BBC.
In a statement, BBC News said: "Along with other media, BBC News was briefed by Kensington Palace on the announcement this afternoon."
There have been calls for privacy from the palace after weeks of speculation and conspiracy theories about the royal couple.
This had intensified after the withdrawal by photo agencies of a photograph of the princess for Mother's Day, on 10 March, because of concerns over digital alterations, for which the princess subsequently apologised.
There were also social media theories over video footage showing Prince William and Catherine shopping in Windsor earlier this week.
Leaders react
Prime Minister Rishi Sunak said Catherine had shown "tremendous bravery" with her statement, wishing her a "speedy recovery".
"In recent weeks she has been subjected to intense scrutiny and has been unfairly treated by certain sections of the media around the world and on social media," he said.
"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, adding he was "heartened" by Catherine's "optimistic tone and her message of faith and hope".
"Any cancer diagnosis is shocking. But I can only imagine the added stress of receiving that news amid the lurid speculation we've seen in recent weeks."
William and Catherine are "entitled to privacy and, like any parents, will have waited to choose the right moment to tell their children".
Professor Pat Price, an oncologist and founder of the Catch Up With Cancer campaign, praised the princess for speaking out "so openly and frankly".
"As was the case with the King's diagnosis, her doing so will no doubt give many more people that vital prompt to get themselves checked and give others affected by cancer more confidence to speak about their own experience.
"This announcement is a stark and shocking reminder that cancer is no respecter of age or social status."
- Additional reporting by Jemma Crew, BBC
- This story was first published by the BBC.