  We analyse in detail the $SL(2,R)$ black hole by extending standard
techniques of Kac-Moody current algebra to the non-compact case. We construct
the elements of the ground ring and exhibit W-infinity type structure in the
fusion algebra of the discrete states. As a consequence, we can identify some
of the exactly marginal deformations of the black hole. We show that these
deformations alter not only the spacetime metric but also turn on non-trivial
backgrounds for the tachyon and all of the massive modes of the string.
