The asymptotic equation derived in , φτ+∇4φ+∇2[(1- 1 2φ)φ]+αφ=0, to describe the interfacial structure of the solidification front of a dilute binary alloy in the limit in which the solute rejection coefficient is close to unity, is demonstrated to be also valid to lowest order when the thermal diffusivities in the liquid and in the solid are unequal. At higher order nonequal thermal diffusivity effects contribute an integral term. A uniformly valid approximation for the interfacial structure is obtained. The integral term is shown to be capable of producing slowly travelling waves or breathing solutions along the interfacial front when the thermal diffusivity of the liquid is sufficiently larger than the thermal diffusivity of the solid. This slowly oscillatory behavior may signal the proximity of a chaotic region located within the region of linear instability. Latent heat of fusion effects are also discussed.