The pair got engaged earlier this month and are set to tie the knot in spring 2018.
In a statement, 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.
"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
"The couple will live in Nottingham Cottage at Kensington Palace."
Kensington Palace said the newly-engaged couple will be appearing for a photo call there later this afternoon.
There were rumours that 33-year-old Harry spoke to the Queen in the summer while at Balmoral to ask for the opportunity to introduce Markle.
It was also reported that the couple took tea with the Queen at Buckingham Palace in October, which some suggested may have been used as an opportunity to ask for her blessing for an engagement.
Markle has already met the Prince of Wales and Duchess of Cornwall, who are reportedly fans.
The 36-year-old actor, who has been married once before, is best known for her role in television legal drama Suits.
Markle spoke publicly for the first time about the relationship in September, when she told Vanity Fair: "We're two people who are really happy and in love.
"We were very quietly dating for about six months before it became news, and I was working during that whole time, and the only thing that changed was people's perception."