Harry, fifth in line to the throne, will marry Ms Markle 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”.
Harry, 33 and Ms Markle, 36, made their first public appearance as a couple in September.
Their 16-month relationship began in July 2016 when they met through mutual friends in London.