Gerald Cavendish Grosvenor, a close friend of the Royal family, died on Tuesday afternoon, having suddenly become ill.
He had been transferred from his Abbeystead Estate to the Royal Preston Hospital in Lancashire.
The Prince of Wales and Duchess of Cornwall are "deeply shocked and greatly saddened" by the sudden death, a Clarence House spokeswoman said.
The Queen and Duke of Edinburgh will send a message of condolence to the Grosvenor family.
One of the Duke's four children, his only son Hugh, is Prince George's youngest godfather, while his wife Natalia is godmother to the Duke of Cambridge.
The landowner was said to be worth around 10.8 billion dollars (Â£8.3 billion), according to Forbes, making him the 68th richest billionaire in the world, and third in the UK.
He owned 190 acres in Belgravia, adjacent to Buckingham Palace and one of London's most expensive areas, as well as thousands of acres in Scotland and Spain.
The Grosvenor family's spokeswoman said on Tuesday: "It is with the greatest sadness that we can confirm that the Duke of Westminster, Gerald Cavendish Grosvenor (64) died this afternoon at Royal Preston Hospital. He was taken there from the Abbeystead Estate in Lancashire where he had suddenly been taken ill.
"His family are all aware and they ask for privacy and understanding at this very difficult time.
"No further comment will be made for the time being but further information will follow in due course."
A Buckingham Palace spokeswoman said: "I can confirm that Her Majesty the Queen is aware of the news about the Duke of Westminster. A private message of condolence is being sent by the Queen and the Duke of Edinburgh."
The Duke suffered a nervous breakdown and depression in 1998, saying the pressures of business and the great number of public appearances he was making overcame him.
In his early 20s, on becoming trustee of the estate, he had been forced to abandon his dream of a career in the Armed Forces, satisfying his love of all things military by serving in the Territorial Army.
He became the sixth Duke of Westminster at 27, and later credited himself with using his vast wealth responsibly.
He supported a number of charities and good causes, including making a Â£500,000 donation to farmers during the 2001 foot-and-mouth outbreak.
Of his wealth he once said: "Given the choice, I would rather not have been born wealthy, but I never think of giving it up. I can't sell. It doesn't belong to me."