The qualitative features of Ginsparg-Wilson fermions, as formulated by Neuberger, coupled to two-dimensional U(1) gauge theory are studied. The role of the Wilson mass parameter in changing the number of massless flavors in the theory and its connection with the index of the Dirac operator is studied. Although the index of the Dirac operator is not related to the geometric definition of the topological charge for strong couplings, the two start to agree as soon as one goes to moderately weak couplings. This produces the desired singularity in the quenched chiral condensate which appears to be very difficult to reproduce with staggered fermions. The fermion determinant removes the singularity and reproduces the known chiral condensate and the meson mass within understandable errors. ©1999 The American Physical Society.