  We formulate chiral gauge theories non-perturbatively, using two different
cuttoffs for the fermions and gauge bosons. We use a lattice with spacing $b$
to regulate the gauge fields in standard fashion, while computing the chiral
fermion determinant on a finer lattice with spacing $f << b$. This determinant
is computed in the background of $f$-lattice gauge fields, obtained by
gauge-covariantly interpolating $b$-lattice gauge fields. The notorious
doublers that plague lattice theories containing fermions are decoupled by the
addition of a Wilson term. In chiral theories such a term breaks gauge
invariance explicitly. However, the advantage of the two-cutoff regulator is
that gauge invariance can be restored to $O(f^2/b^2)$ by a {\it one-loop}
subtraction of calculable local gauge field counterterms. We show that the only
obstruction to this procedure is the presence of an uncancelled gauge anomaly
among the fermion representations. We conclude that for practical purposes, it
suffices to choose $f/b \sim b/L$, where $L^4$ is the physical volume of the
system. In our construction it is simple to prove the Adler-Bardeen theorem for
anomalies in global currents to all orders. The related subject of fermion
number violation is also studied. Finally, we discuss the prospects for
improving the efficiency of our algorithm.
