The owners are currently looking to relocate the Cinema and Bar to another venue, but no alternatives have been announced as yet.
The members club opened in 2003 and featured a 75-seat cinema and a 1930s-style lounge bar with a capacity for 120.
As well as being a popular venue for corporate events, Rex was a favourite of the film, media and music set and was frequented by celebrities such as Jude Law, Girls Aloud and Daniel Craig.
Other popular celebrity haunt, Chinawhite, which closed last month, is still looking for a new home.