  We study supercurrent conservation for the four-dimensional Wess-Zumino model
formulated on the lattice. The formulation is one that has been discussed
several times, and uses Ginsparg-Wilson fermions of the overlap (Neuberger)
variety, together with an auxiliary fermion (plus superpartners), such that a
lattice version of U(1)_R symmetry is exactly preserved in the limit of
vanishing bare mass. We show that the almost naive supercurrent is conserved at
one loop. By contrast we find that this is not true for Wilson fermions and a
canonical scalar action. We provide nonperturbative evidence for the
nonconservation of the supercurrent in Monte Carlo simulations.
