  String propagation on a cone with deficit angle $2\pi (1- \frac{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 ${\scriptstyle R} \rightarrow {1\over R} $ duality.
