According to the Daily Record, Andrew has given up his honorary membership at the club in light of the Epstein scandal.
The course and club are also to come under the management of the St Andrews Links Trust on a long-term lease.
In addition to the name, the club's logo will also be modified and remove its royal association. In place of the lion rampant holding a saltire flag, the logo will soon bear an illustration featuring trees.
The Duke’s Course will continue to be run by golf resort operator Kohler, alongside its ownership of the Old Course Hotel, until January 5.
Neil Coulson, who is the chief executive of St Andrews Links Trust, said: "The agreement for The Craigtoun Course will see the first new course added to the Home of Golf’s portfolio in 18 years and presents an ideal opportunity to expand the golf experiences we offer.
"We are delighted to have reached agreement with the Old Course Hotel and look forward to taking over what it is already a significant asset for St Andrews and Scotland and building on its reputation.”
Andrew will also soon leave Royal Lodge, the 30-room property he has lived in since 2004. He is reportedly set to move into a property on the Sandringham Estate, whilst his ex-wife, Sarah Ferguson, will also move out of Royal Lodge.