  We present a method for formulating gauge theories of chiral fermions in
lattice field theory. The method makes use of a Wilson mass to remove doublers.
Gauge invariance is then restored by modifying the theory in two ways: the
magnitude of the fermion determinant is replaced with the square root of the
determinant for a fermion with vector-like couplings to the gauge field; a
double limit is taken, in which the lattice spacing associated with the fermion
field is sent to zero before the lattice spacing associated with the gauge
field. The method applies only to theories whose fermions are in an
anomaly-free representation of the gauge group. We also present a related
technique for computing matrix elements of operators involving fermion fields.
Although the analyses of these methods are couched in weak-coupling
perturbation theory, it is argued that the computational prescriptions are
gauge invariant in the presence of a nonperturbative gauge-field configuration.
