Strings on a Cone and Black Hole Entropy

String propagation on a cone with deficit angle 2π(1 − 1 N) is described by constructing a non-compact orbifold of a plane by a Z N subgroup of rotations. It is modular invariant and has tachyons in the twisted sectors that are localized to the tip of the cone. A possible connection with the quantum corrections to the black hole entropy is outlined. The entropy computed by analytically continuing in N would receive contribution only from the twisted sectors and be naturally proportional to the area of the event horizon. Evidence is presented for a new duality for these orbifolds similar to the R → 1 R duality.