The prince, fifth in line to the throne, will marry Markle next spring and the pair will live at Nottingham Cottage at Kensington Palace in London.
A statement issued by Clarence House said: "His Royal Highness The Prince of Wales is delighted to announce the engagement of Prince Harry to Ms Meghan Markle.
"The wedding will take place in Spring 2018. Further details about the wedding day will be announced in due course.
"His Royal Highness and Ms Markle became engaged in London earlier this month. Prince Harry has informed Her Majesty The Queen and other close members of his family. Prince Harry has also sought and received the blessing of Ms Markle's parents."
The couple have been dating since last summer and the 36-year-old actor is rumoured to be moving to the UK after quitting the hit US show Suits.
In a statement, Prince Harry said he was "delighted to announce" the engagement and had gained the blessing of Ms Markle's parents.
Kensington Palace said Prince Harry and Markle would appear for a photo call at Kensington Palace later on Monday afternoon.
The couple will also take part in a broadcast interview to be broadcast this evening UK time.
In a statement issued by Kensington Palace, Ms Markle's parents Thomas Markle and Doria Ragland celebrated news of the engagement.
"We are incredibly happy for Meghan and Harry. Our daughter has always been a kind and loving person," they said.
"To see her union with Harry, who shares the same qualities, is a source of great joy for us as parents. We wish them a lifetime of happiness and are very excited for their future together."
The Queen gave her blessing to the marriage as Prince Harry is obliged to ask her permission under the Succession to the Crown Act 2013.
Rumours about the pair's engagement have abounded in recent weeks and speculation of a major royal announcement has mounted.
The prince and Markle made their first public appearance as a couple this September at the Invictus Games.
Their 16-month relationship began after the couple met through mutual friends in London but they have grown particularly close in recent months. Markle has been reported to be staying regularly at the prince's home at Kensington Palace.
The Canada-based actor, who rose to fame for playing ambitious paralegal Rachel Zane on Suits, has been married once before. Markle married film producer Trevor Engelson in Jamaica in 2011 but they broke up after just two years.
News of the engagement is set to spark a media circus that is likely to put yet more strain on Harry's already fraught relationship with the press. After their relationship became public just over a year ago, the 33-year-old royal condemned the paparazzi for subjecting Markle to a “wave of abuse and harassment”.
In a statement, the Palace attacked the “racial undertones” of media coverage of Markle and said the prince feared for her safety. The actress, whose father is white and mother is African American, has described herself as biracial and “half black and half white” on her website.
“My dad is Caucasian and my mom is African-American. To say who I am, to share where I’m from, to voice my pride in being a strong, confident mixed-race woman," Markle has previously said.
Markle, who has criticised President Donald Trump, is a humanitarian and gender equality campaigner. She is a UN women’s advocate (for Political Participation and Leadership) and has publicly supported Emma Watson’s He For She campaign.
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