The 77-year-old royal was taken to an undisclosed London hospital in the early hours of Monday where he was assessed by doctors.
A Buckingham Palace spokesman said: "He was taken ill in the early hours of Monday and admitted to hospital for assessment."
He added that medical staff "diagnosed him as having had a mild stroke".
The Buckingham Palace spokesman added: "He's still being assessed and it's not thought to be serious.
"My understanding is he's feeling well and looking forward to resuming official engagements as soon as possible."
It is thought the Duke, whose first name is Edward, is still being treated at a London hospital.
It is understood the Queen, who herself has been ill recently with the symptoms of gastroenteritis, has been informed.
The Duke is president of the Stroke Association and its chief executive Jon Barrick wished him well: "We are deeply saddened to hear about His Royal Highness the Duke of Kent's stroke.
"Our thoughts are with him and his family and we wish him a very speedy recovery.
"The Duke of Kent has been president of the Stroke Association since our inception over 20 years ago and he has shown tremendous support and dedication to our cause.
"He has hosted a number of high profile events and opened our Life After Stroke Centre in Bromsgrove last year."
The Duke inherited his royal title at the age of six after his father's life was tragically cut short in a plane crash in 1942.
After a lengthy military career, he has spent many years performing royal duties on behalf of his cousin the Queen.
He is perhaps best known for being President of the All England Lawn Tennis and Croquet Club and presenting the Wimbledon trophies each summer.
The royal is also the country's top Freemason - the secret society's Grand Master of the United Grand Lodge of England.
"Eddie" Kent was a respected Army officer who also acted as Britain's international trade ambassador for 25 years.
His wife the Duchess of Kent has withdrawn from public life, dropping her HRH title, after suffering from depression and other illnesses.
She now works as a music teacher, preferring to be known as plain "Katharine Kent".
The couple have three children George, the Earl of St Andrews, Lady Helen Taylor, and Lord Nicholas Windsor.
A stroke is when the blood supply to part of the brain is cut off - either by a blood clot or bleeding - and brain cells are damaged or die.
But the Duke could have suffered a transient ischaemic attack, also known as a mini stroke, where there is only a momentary lack of blood flow to the brain.
If he has experienced the lesser illness the royal could be back attending official engagements within a few weeks.
His next royal event is on April 16 when he is scheduled to visit West Sussex to tour a number of places including a school and a museum.