The much-anticipated news was confirmed by the Prince of Wales in a brief statement issued by Clarence House on Monday morning.
Their wedding will take place in Spring 2018 and they will live in Nottingham Cottage at Kensington Palace.
They became engaged in London earlier this month.
Welcoming the news, Justin Welby said: "I am absolutely delighted to hear the news that Prince Harry and Meghan Markle are now engaged.
"I have met Prince Harry on a number of occasions and have always been struck by his commitment and passion for his charities, and his immense love for his family.
"Marriage is a special and joyous commitment, one that Jesus celebrated together with friends at the wedding in Cana. I am so happy that Prince Harry and Ms Markle have chosen to make their vows before God.
"I wish them many years of love, happiness and fulfilment and ask that God blesses them throughout their married life together."
Reports have suggested Justin Welby has no objections marrying the pair, despite Markle being a divorcee.
Westminster Abbey has also confirmed it would be happy to host the event.