The operator, previously known as The Draco Pub Company, plans to run the 16-bedroom property in the centre of the market town in its current form before refurbishing early next year.
Stay Original managing director Rob Greacen said: “We are delighted to be the new owners of this historic and much loved local landmark. It’s an exciting addition to our expanding portfolio. We’re going to settle in, treat customers to a very special Christmas and then in the New Year we’ll start to make some improvements to the property.
"At Stay Original we pride ourselves on our food and first-rate service and customers in Shaftesbury can look forward to enjoying the very best of both of these immediately."
The Grosvenor Arms, which has been the site of an inn since medieval times, joins the company's four other pubs with rooms - The Swan in Wedmore, The White Hart in Somerton, Somerset, Timbrell's Yard in Bradford on Avon in Wiltshire and The King's Arms in Dorchester, Dorset. The kitchens of all pubs are overseen by former River Cottage head chef Tom Blake.
The 37-bedroom former Best Western hotel The King's Arms, was bought out of receivership in late 2015 by Stay Original. The 300-year-old hotel, once frequented by author Thomas Hardy, is currently undergoing a major refurbishment before re-opening next year.