  We investigate a supersymmetric theory with an extra U(1) gauge symmetry
surviving down to low energies. The extra U(1) is assumed to originate from an
E$_6$ grand unified theory (GUT). We show that if one assumes universal soft
supersymmetry breaking scalar masses at the GUT scale, and requires the mass of
the additional U(1) gauge boson to satisfy phenomenological bounds, then the
conditions for electroweak symmetry breaking will provide stringent
restrictions on the allowed parameter space of such a theory. We also determine
the masses of standard as well as exotic sfermions and find that it is possible
for the latter to be lighter than the former. An interesting specific
observation is that a light stop is difficult to accommodate with an extra U(1)
gauge symmetry.
