The message accompanying the post, which references the star's account, says simply: "Coming soon... @teddysphotos #WMHD."
The hashtag for World Mental Health Day is #WMHD.
In the brief footage, lasting just ten seconds, Sheeran approaches a door, rings a doorbell, which plays God Save The Queen, to be greeted by Harry, who says: "Hello mate."
In arguably a meeting of the two-most famous red-haired people in the country, the royal also tells the singer: "It's like looking in the mirror."
The singer laughs, shakes the duke's hand and replies: "Hey mate. How you doing? Nice to see you. Do you mind if I bring the camera in?"
Harry replies: "Of course. Come on in guys."
It finishes with a black screen with the words: "Tomorrow 10th October."
Within minutes of being posted, the video had been viewed more than 625,000 times.
Among those responding to the clip, was one poster who wrote: "The two favourite gingers of England. Perfect.”
The advert, part of the NHS and Public Health England's Every Mind Matters initiative, aims to educate people about identifying the early signs of mental health issues, as well as providing simple steps for dealing with stress, improve sleep and mood.