Hugh Grosvenor is the only son of Gerald Cavendish Grosvenor, who died on Tuesday afternoon after being transferred from his Abbeystead Estate to the Royal Preston Hospital in Lancashire.
A spokeswoman for the family said the cause of death is not yet known but police said the duke became ill while walking in the Trough of Bowland, a local beauty spot, and there are no suspicious circumstances.
The 64-year-old father of four was said to be worth around 10.8 billion dollars (Â£8.3 billion), according to Forbes, making him the 68th richest person in the world, and third in the UK.
The Duke and Duchess of Cambridge, whose son Prince George is godson to Hugh Grosvenor, were "very sad" to hear of the death, Kensington Palace said in a statement.
It added: "Their thoughts are very much with his family this morning."
Close friends the Prince of Wales and Duchess of Cornwall were "deeply shocked and greatly saddened" by the duke's sudden death, a Clarence House spokeswoman said.
The Queen and Duke of Edinburgh will also send a private message of condolence to the Grosvenor family.
Sir Gerald 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 title and the land will pass to Hugh Grosvenor - who is two years younger than his father was when he took on the fortune at the age of 27 as the sixth duke.
A staunch supporter of a number of charities and good causes, the duke credited himself with using his vast wealth responsibly.
This included making a Â£500,000 donation to farmers during the 2001 foot-and-mouth outbreak, as well as fighting a legal case against Westminster City Council in 1990, centred on a number of social housing flats built on the family's land in Pimlico, London.
The buildings were designed by architect Sir Edwin Lutyens and built between 1928 and 1930. Assigning the lease of the flats to the council for 999 years in 1937, the second duke stipulated they must be used as housing for the working classes.
When the council wanted to sell the properties below the market value to those working in the borough, the duke refused.
In court, the authority argued that the working classes no longer existed, but the judge ruled in the duke's favour, backing the clause and his bid to keep low-cost accommodation.
Of his wealth, the duke 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."
In his early 20s, on becoming trustee of the estate, he was forced to abandon his dream of a career in the armed forces, but satisfied his love of all things military by serving in the Territorial Army.
Paying tribute, the Soldiers' Charity, which the duke supported, said he had been involved in the "championing of soldiers and veterans" and "will be very fondly remembered".
He 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.
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."
He is survived by his wife, Natalia Phillips, who he married in 1978 and their daughters, Lady Tamara, born in 1979, Lady Edwina in 1981 and Lady Viola in 1992.
The duke was on his Abbeystead Estate in Lancashire - described as an "area of outstanding beauty" with "rolling hills" by the Grosvenor Estate website - when he was taken unwell.
A spokeswoman for Lancashire Police told the Press Association officers were called at around 5pm on Tuesday and made aware of the death of a 64-year-old man at Royal Preston Hospital.
"He was airlifted to hospital after he had been taken ill whilst walking in the Trough of Bowland. There are no suspicious circumstances and a file will be passed to the coroner," she added.