We formulate chiral gauge theories non-perturbatively, using two different cutoffs 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(f2/b2) by a 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 ∼ b/L, where L 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.