I did once have a beer in the garden when it was The Spring Gardens. It became La Demi Lune for a while, from what I remember, then shut down. Having just Googled it, before it was the Spring Gardens apparently it was "Durty Nelly's"!
From what I can see on Street View it's just flats now, but I can't be sure without going and having a proper look around the front. Maybe next time...
GPS Coordinates:
51.45029, -2.61581
Location: Hotwells East