The prince, who was married to the Queen’s sister Princess Benedikte, died suddenly on Monday evening at his Berleburg Castle home in Germany, reports Danish news agency Ritzau.
“It is with great sorrow that I, on behalf of Her Royal Highness Princess Benedikte, must announce that His Highness Prince Richard has passed away,” the princess’ private secretary wrote in a statement.
The prince and princess had been married since 1968.
The princess has returned to Germany after a short stay in Denmark, reports the news agency.
Prince Richard had a number of health problems in recent years, and was treated for skin cancer in Germany in 2003 before later being diagnosed with a malignant lump in his thigh.
The royal pair have three children, Gustav, Alexandra and Nathalie, and four grandchildren.
Gustav, the only son, will inherit the late prince’s castle and 15,000 hectares of forest grounds.
The father and son had a close relationship which was strengthened during Prince Richard’s cancer treatment, reports Ritzau.
Prince Richard, full name Richard Casimir Karl August Konstantin zu Sayn-Wittgenstein-Berleburg, is the son of Swedish-born duchess Margareta, who was friends with Queen Margrethe and Princess Benedikte’s mother, Queen consort Ingrid.
The prince’s father, Prince Gustav Albrecht, was a captain in the German army and was killed on the eastern front during the Second World war.
After the war, Margareta returned to Sweden, where the young Richard lived until returning to Germany to study law as well as forestry.