  The complex-field zeros of the Random Energy Model are analytically
determined. For T<T_c they are distributed in the whole complex plane with a
density that decays very fast with the real component of H. For T>T_c a region
is found which is free of zeros and is associated to the paramagnetic phase of
the system. This region is separated from the former by a line of zeros. The
real axis is found to be enclosed in a cloud of zeros in the whole frozen phase
of the system.
