I'm not sure if they're one and the same thing but I have always thought it was the Reading Room at the British Museum (i.e. that the British Library was located at the Museum at that time)
The Reading Room at the British Museum was, until the British Library was finally completed, the store of the national book collection. It was where Marx did much of his research.
Since the 1990's when the new building opened at St Pancras, the British Library has been its own building.