During the most luminous portion of the asymptotic giant-branch phase, models of intermediate-mass stars first become carbon stars and then produce s-process isotopes in the solar-system distribution. Recent observations of the optically most luminous carbon stars in the Magellanic Clouds introduce the possibility that real intermediate-mass stars lose their hydrogen-rich envelopes during the asymptotic giant-branch phase before they have made s-process isotopes both in large quantities and in the solar system distribution. This encourages a search for alternate sources of these isotopes. A promising site for the production of some neutron-rich isotopes isthe convective helium-carbon region that appears in accreting white dwarfs during helium shell flashes. For appropriate accretion rates, overlap of matter in successive convective zones may lead to an exponential distribution of exposures. Further, because of a small entropy barrier between the convective shell and the hydrogen-rich envelope, protons enter the shell and provide a source of neutrons that, for appropriate accretion rates, is repetitive in strength and either dominates or is complementary to the /sup 22/Ne(..cap alpha..,n)/sup 25/Mg source. This permits an estimate of the distribution of neutron-rich isotopes that is formed after many flashes. The distribution, in most instances, tends to be weighted more toward heaviermore » elements than is the case when /sup 22/Ne(..cap alpha.., n)/sup 25/Mg is the sole source of neutrons. Hence, accreting white dwarfs cannot be major contributors to the enrichment of the interstellar medium in most s-process isotopes. Considerable effort should be devoted toward demonstrating whether or not the bolometrically most lumious asymptotic giant branch stars in local systems obey M/sub BOL/ /sup min/ -6.5, then either the source of most Galactic s-process isotopes is as yet unknown, or the rate of the /sup 22/Ne(..cap alpha.., n)/sup 25/Mg reaction has been considerably underestimated.« less