The stability of a horizontal thermal boundary layer embedded within a very viscous fluid is investigated using the formalism of linear stability analysis. Thin thermal boundary layers in deep fluid regions and in the absence of phase transition and dynamical effects are thereby shown to be unstable at extremely long wavelengths. The stability of the internal thermal boundary layer which may exist at 660 km depth in the Earth's mantle as a consequence of the dynamical influence of the endothermic phase transition from &ggr; spinel to a mixture of perovskite and magnesiow¿stite, recently discussed in some detail by Solheim and Peltier <1994a>, is investigated in order to better understand the avalanche effect observed in this and similar nonlinear, time dependent simulations of the mantle convection process. It is demonstrated that if the stability problem is treated as purely thermal, then the boundary layer is predicted to be extremely unstable and the presence of the 660-km endothermic phase transition at middepth within the boundary layer is further destabilizing. When the kinematic effect of flow convergence onto the boundary layer and phase transition region is active, however, it is shown that the layer may be strongly stabilized. In the regime of physically realistic velocity convergence, the critical Rayleigh number is predicted to lie in the range suggested by the numerical simulations of Solheim and Peltier <1994a>. A threshold value of the magnitude of the Clapeyron slope of the endothermic phase transition for a given velocity convergence is also shown to exist, beyond which the fastest-growing mode of instability changes from avalanche type to layered type.¿ 1997 American Geophysical Union