  It is shown that U(1) chiral gauge theories with anomaly-free multiplets of
Weyl fermions can be put on the lattice without breaking the gauge invariance
or violating any other fundamental principle. The Ginsparg-Wilson relation
plays a key role in this construction, which is non-perturbative and includes
all topological sectors of the theory in finite volume. In particular, the
cancellation of the gauge anomaly and the absence of global topological
obstructions can be established on the basis of this relation and the lattice
symmetries alone.
