Moss Bros said it sold the business to concentrate on its core branded suit business. Earlier this year it also offloaded 15 Hugo Boss stores.
JD said it bought the business and assets of eight Cecil Gee stores in cash, as well as the rights to use the Cecil Gee name.
Peter Cowgill, executive chairman of JD, said acquiring the Cecil Gee brand gave the company a "springboard" to develop a "well-funded profitable premium brand fashion chain".
JD will run the Cecil Gee brand as a separate entity. It also operates two premium chains, Bank and Scotts.
The Cecil Gee business had been made up of nine menswear retail stores. One store in Glasgow that is not part of the sale will be converted to a new format Moss Bros store in the next few weeks.
Brain Brick, chief executive of Moss Bros, said: "Although we are ever mindful of the fragile trading conditions in the UK, we are making good progress towards leadership of the attractive niche which we have as the UK's number one branded suite specialist."