Harry, fifth in line to the throne, will marry Meghan Markle, star actress of the American TV series, Suits next spring and will live at Nottingham Cottage at Kensington Palace in London.
The couple, who have been dating since the summer of 2016, secretly got engaged earlier this month.
In a statement, Prince Charles said he was “delighted to announce” his son’s news and that Harry had received the blessing of Ms Markle’s parents.
Only the Queen and “other close members of his family” initially knew of the engagement, which took place in London.
The announcement, issued by Clarence House, the Prince of Wales and Duchess of Cornwall’s official residence, said details about the wedding day would be “announced in due course”.
The engaged couple will appear for photographs outside Kensington Palace on Monday afternoon, and will take part in a broadcast interview in the evening.
The Queen and Duke of Edinburgh said they were “delighted for the couple and wish them every happiness”, a Buckingham Palace spokesman said.
Prince William and Catherine, Duchess of Cambridge, said they were “very excited for Harry and Meghan”, adding: “It has been wonderful getting to know Meghan and to see how happy she and Harry are together.”
Prime Minister Theresa May offered her “very warmest congratulations” and wished the couple “great happiness for the future”.
Labour leader Jeremy Corbyn said: “I wish them well, I hope they have a great life together”.
He joked: “Having met Harry a couple of times I’m sure they’re going to have a great deal of fun together.”
Ms Markle’s parents, Thomas Markle and Doria Ragland, said they were “incredibly happy”, adding: “To see her union with Harry, who shares the same qualities, is a source of great joy for us as parents”.
Harry, 33 and Ms Markle, 36, made their first public appearance as a couple in September.
They were seen together at the opening ceremony of the Invictus Games, an international sporting event set up by the prince for injured soldiers.
Their 16-month relationship began in July 2016 when they met through mutual friends in London.
Harry first confirmed his relationship with Ms Markle in November 2016, in a statement from Kensington Palace attacking the media for subjecting her to a “wave of abuse and harassment”.
That statement said the couple were “a few months into a relationship” and it was “not right” that Ms Markle should receive such treatment.
It described nightly legal battles to keep defamatory stories out of papers, attempts by reporters and photographers to get into her home and the “bombardment” of nearly every friend and loved one in her life.