The video, which is on YouTube is called ‘Andy and Fergie- before they married’, shows the divorced parents of Princess Eugenie and Princess Beatrice laughing together.
The footage has surfaced just weeks before the pairs’ second daughter Princess Eugenie is due to wed her long term boyfriend Jack Brooksbank in Windsor.
Prince Andrew is dressed in his pilot uniform in the video and the Duchess asks him about flying and simulators.
Fergie looks radiant in a long white dress and her ginger locks down.
After the Duchess says the nickname, Prince Andrew said: “I thought we were going to keep that name secret.
To which Ms Ferguson replied to the laughing prince: “I haven’t just used it.”
The name is difficult to hear but sounds like ‘Bunny’.
At one point during the video, the Duchess asks the Prince to kiss her saying: “Hurry up and kiss me, they are filming.
Before reaching forward and kissing Ms Ferguson, Prince Andrew says jovially: “You are a monster.”
He tells the people filming that he won’t kiss her until the balcony, referring to the traditional kiss a royal bride and groom make on the balcony of Buckingham Palace when they get married.
The footage shows the pair playing around, flirting and they appear to be very much in love.
The video has been viewed more than 140,000 times on YouTube.
It also shows the young couple in a shop buying luxury plates and glasses.
The Duchess and Prince Andrew are rumoured to still live together in the Royal Lodge in Windsor and she still is close to the Royal Family and was invited to the wedding of Prince Harry and Meghan Markle.
The property was previously the private residence of the Queen’s Mother for over 70 years, before being gifted to Prince Andrew in 2003.
The house also has gardens, a private chapel, 21 acres of secluded grounds and eight separate additional properties for staff.
Prince Andrew served as an active-duty helicopter pilot in the Royal Navy.
The Duchess has recently described her 1986 wedding to Prince Andrew as “the greatest day of her life” and she has revealed that she still keeps a video copy of her wedding.
After a turbulent marriage, they divorced in 1996 when the Duchess received a settlement fee of £3 million.