An investigation is made of the stability of the shape of a moving planar interface between the solid and liquid phases during the solidification of a pure substance. The prototype problem of controlled two-dimensional growth of a pure solid into a thermally undercooled liquid bath is considered. The model employed postulates diffusion of heat with equal thermal diffusivities in both phases under the simplifying assumptions that these phases are infinite in extent and there is no convection in the liquid phase. In addition, modified versions of the conservation of heat boundary condition and the Gibbs—Thomson equation, developed by Wollkind and Maurer in earlier work, are imposed at the solid—liquid interface. The main results of our non-linear analysis, which can be represented in a plot of undercooling versus solidification speed are that the interface can be unstable to finite amplitude disturbances in some regions where linear theory predicts stability to infinitesimal disturbances and that it can exhibit finite amplitude equilibrium for other ranges of parameter values. These results are interpreted in relation to the growth and structure of dendrites.