Seven photographs taken in the Norfolk countryside by photographer Josh Olins will appear in the June edition.
In her first sitting for a magazine, the duchess is seen looking relaxed and wearing casual clothes.
The shoot was in collaboration with the National Portrait Gallery, where two images are being displayed in the Vogue 100: A Century of Style exhibition.
The duchess, who has a keen interest in photography, has been patron of the National Portrait Gallery since 2012.
Nicholas Cullinan, director of the National Portrait Gallery, said: "Josh has captured the duchess exactly as she is - full of life, with a great sense of humour, thoughtful and intelligent, and in fact, very beautiful."
He said the images also encapsulated what Vogue had done over the past 100 years - "to pair the best photographers with the great personalities of the day, in order to reflect broader shifts in culture and society".
Alexandra Shulman, editor-in-chief of British Vogue, said: "To be able to publish a photographic shoot with the Duchess of Cambridge has been one of my greatest ambitions for the magazine."
The collaboration for the June edition had resulted in "a true celebration of our centenary as well as a fitting tribute to a young woman whose interest in both photography and the countryside is well known", she said.
Other royal portraits to have featured in the fashion magazine include Diana, Princess of Wales - who graced the cover four times - and Princess Anne.
The duchess is to visit the exhibition at the National Portrait Gallery on Wednesday, Kensington Palace said.

Ministers are urging people, especially those from ethnic minorities, to sign up as a donor as their new year's resolution for 2016.
About 25% of people on the UK transplant waiting list are from black, Asian and minority ethnic communities.
But only 5% of donors are from these communities, making it harder to find suitable organ for transplant.
A transplant is much more likely to be successful if the donor and recipient have the same ethnic origin.
The shortage of suitable organs means people from minority ethnic communities wait, on average, twice as long as the general population for organ transplants.
Muhammad Azam, 43, waited two months on the organ donor register before his life-saving liver transplant in December 2014.
The father-of-two, from Edinburgh, was diagnosed with Hepatitis C in 2006 and in 2012 he was told his chance of survival relied on a liver transplant.
He said: "I was constantly exhausted and was even unable to walk at times because I felt so weak. It was a miserable existence."
Since his transplant, Mr Azam has been able to return to work and his life has completely transformed.
He said: "After the operation I just felt instantly better than I had in months.
"Since then my recovery has been fantastic. Two days after the transplant I was able to walk around again and within nine days I was allowed to go home to recuperate.
"I think there can be misconceptions surrounding organ donation and religion, but actually all major religions in the UK support it in principle.
"I'd encourage anyone who hasn't done so already to join the NHS organ donor register. It's the most generous thing anyone could ever do for another person.
"It really is the greatest gift of all."
A three-year programme delivered by the Scottish government and Kidney Research UK has recruited "peer educators" from black, Asian and minority ethnic communities who act as champions for organ donations within their communities.
Public Health Minister Maureen Watt said: "It is a reality that we need more donors from black, Asian and minority ethnic communities if we are to be able to help people like Muhammad get a new lease of life."

Pending a successful medical, the 20-year-old Colombian would become the sixth signing of the summer at Ibrox.
Morelos spent 16 months in Finland, after arriving from Independiente Medellin, and scored 47 goals in 62 appearances for HJK.
The forward, who has six Under-20 caps, would require a work permit to play in the UK.
Morelos would join Bruno Alves, Ryan Jack, Dalcio, Fabio Cardoso and Daniel Candeias as new arrivals at Ibrox, as manager Pedro Caixinha looks to overhaul his squad in time for the Europa League qualifiers, which begin on 29 June.
The Duchess of Cambridge will feature on the cover of British Vogue to mark the magazine's centenary.
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A plea has been made to give others "a second chance at life" by registering to become an organ donor.
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HJK Helsinki striker Alfredo Morelos arrived in Edinburgh this morning ahead of his expected move to Rangers.