In order to explore the influence of a specific type of defect on the phenomenon of morphological instability, we have calculated the time-dependent shape of a nearly planar interface, intersected perpendicularly by a grain boundary, during solidification of a pure substance at constant velocity. The calculational methods and principal assumptions are similar to those employed in previous theories of morphological stability except that the slope of the interface is maintained at a finite and constant value, s, in the immediate vicinity of the grain boundary groove. The position of the solid-liquid interface is described by the equation y = W( x,t) where t is the time and W( x, 0) → 0 as ‖ x‖→∞ (all quantities are assumed independent of z). Whereas the stability-instability criterion is found to be identical to that for an interface without a grain boundary, the boundary is found to be an effective initial perturbation. Under conditions for instability, the depth of the grain boundary groove increases exponentially with time and an oscillatory instability propagates laterally from the boundary. Under conditions for stability, the interface eventually attains a time-independent shape given by W( x, t→∞) = (- s/a)exp(- ax), where a 2 = ( G S+ G L)/2 T MΓ, G S and G L are cond ductivity-weighted temperature gradients in solid and liquid, respectively, T M is the melting temperature and Γ is a capillary constant. For conditions corresponding to the demarcation between stability and instability, a mode of thermal grooving, similar to that previously described by Mullins, is found. A meaningful criterion for instability is shown to be the exponential growth of perturbations while, conversely, stability entails their exponential decay; phenomena such as the algebraic increase of amplitude characteristic of thermal grooving are shown to be manifestations of constraints. Finally, the situation where the interface shape is allowed to depend on z is shown to be describable by a superposition of W( x, t) with a function W o( x, z, t) that corresponds to the conventional case where the grain boundary is absent.