Observations suggest a connection between the low magnetic fields of binary and millisecond pulsars and their being processed in binary systems, indicating accretion-induced field decay in such cases. A possible mechanism is that of rapid ohmic decay in the accretion-heated crust. The effect of accretion on purely crustal fields, for which the current loops are completely confined within the solid crust, is two-fold. On the one hand the heating reduces the electrical conductivity and consequently the ohmic decay time-scale, inducing a faster decay of the field. At the same time the material movement, caused by the deposition of matter on top of the crust, pushes the original current-carrying layers into deeper and denser regions where the higher conductivity slows the decay down. This results in a competition between these two opposing processes. The mass of the crust of a neutron star changes very little with a change in the total mass; accretion therefore implies assimilation of the original crust into the superconducting core. When the original current-carrying regions undergo such assimilation, further decay is stopped altogether. We perform model evolutionary calculations for a range of values of the accretion rate and the crustal temperature. We find that in all cases an initial phase of rapid decay is followed by a slow-down and finally a freezing of the surface field. The pre-accretion phase of field decay in the effectively isolated neutron star plays a significant role. In this phase the currents diffuse down through the whole of the crust by pure ohmic dissipation, and the longer it lasts the deeper the currents penetrate. If prior to the accretion phase the currents have already penetrated to the regions of high density and hence high conductivity, the effect of crustal heating is not as dramatic.