We consider the peritectoid reaction α + ß→ ω as it occurs at previously existing planar α/ß interfaces, and at sufficiently low temperatures that volume diffusion is negligible and interphase boundary diffusion is rate-controlling. With the further assumption that the participating phases have fixed compositions and that the interfacial diffusion process is driven by gradients in interfacial curvature, we obtain a unique solution for the growth of the product ω layer along the α/ß interface. Both the layer thickness and the steady lengthening velocity are predicted. The layer thickness is of the same order as the capillary length; it would be difficult to detect by conventional means, but could be revealed for example by T.E.M. The layer growth is determined entirely by the undercooling, the interfacial energies and the relevant kinetic quantities. The presence of such thin layers at parent phase interfaces is expected to exert a profound influence on microstructurally determined properties.