The painting, commissioned last year as a gift for the people of Cambridgeshire, shows the pair standing side-by-side, with Kate in an emerald dress and William in a black suit.
It will go on public display in the university city.
The portrait was painted by award-winning artist Jamie Coreth, described as “one of Britain’s leading portrait artists” who “focuses on the character of his sitters, evoking a sense of their presence in his work" on his website.
The Duke and Duchess, both 40, met Mr Coreth as they viewed the painting of themselves on Thursday.
Kate and William's portrait
Jamie Coreth/Fine Art Commissions
After looking at the painting, William said: “It’s quite big.”
He told Mr Coreth it was “amazing”.
The Duke and Duchess both studied History of Art at St Andrews University, though William later switched to Geography.
William said in a Big Issue Q&A session published this week to mark his 40th birthday: “I studied a bit of Art History at university.
“Had to give it up.
Kate and William greeted school children as they visited the gallery
Joe Giddens
“I kept falling asleep in the lectures. Terrible.
“We did a lot of Renaissance, which was amazing.
“But then once we got into modern art, I started to get a bit dozy.”
The portrait of William and Kate was commissioned by the Cambridgeshire Royal Portrait Fund, held by the Cambridge Community Foundation, as a gift to Cambridgeshire.
Mr Coreth worked to incorporate the city of Cambridge into the portrait by painting the background with the tones and colours of many of the historical stone buildings that are synonymous with it.
The portrait also includes the use of a hexagonal architectural motif which can be seen on buildings across the university city.
Mr Coreth said it was the “most extraordinary privilege of my life to be chosen to paint this picture”.
He added: “I wanted to show Their Royal Highnesses in a manner where they appeared both relaxed and approachable, as well as elegant and dignified.
“As it is the first portrait to depict them together, and specifically during their time as the Duke and Duchess of Cambridge, I wanted the image to evoke a feeling of balance between their public and private lives.
Kate on the couple's trip to Cambridge today
Joe Giddens
“The piece was commissioned as a gift for the people of Cambridgeshire, and I hope they will enjoy it as much as I have enjoyed creating it.”
Members of the public will be able to view the portrait at the Fitzwilliam Museum for an initial period of three years.
After this, the artwork will be exhibited in other community spaces and galleries around Cambridgeshire.
The painting will also be loaned to the National Portrait Gallery for a short time in 2023 to mark the gallery’s reopening.