  $U(1)$ zero modes in the $SL(2,R)_k/U(1)$ and $SU(2)_k/U(1)$ conformal coset
theories, are investigated in conjunction with the string black hole solution.
The angular variable in the Euclidean version, is found to have a double set of
winding. Region III is shown to be $SU(2)_k/U(1)$ where the doubling accounts
for the cut sructure of the parafermionic amplitudes and fits nicely across the
horizon and singularity. The implications for string thermodynamics and
identical particles correlations are discussed.
