The stability of a combustion wave is one of the most important problems in combustion theory. The solution of the problem can be essentially simplified if the characteristic scales of the processes governing the combustion are small in comparison with the characteristic dimensions of the problem. A common approach to the stability problem of a flame front is to treat the flame front or some inner zone (usually it is the chemical reaction zone) as a surface of discontinuity of zero thickness (Refs. [1,2], and references therein). The model with the discontinuity is widely used to solve the flame stability problem of both gaseous [3,4] and condensed fuels [5—8]. However, the stability problem thus formulated with a surface of discontinuity leads to another difficulty, since it requires an additional boundary condition (jump condition at the surface of discontinuity), which in principle cannot be obtained or justified within the scope of the discontinuity model [9,10]. Because of this, the boundary condition at the discontinuity is introduced as a "physically reasonable" boundary condition. The sensitivity of the solution obtained in this way and the accuracy of the assumed boundary condition can be verified only by solving the complete problem with finite thickness and real structure of the fronts [10]. Such an examination can be a rather difficult mathematical problem. For these reasons and because the flame is a typical example of a wide class of the reacting waves in the deflagration regime (ionizing wave, phase transition wave, ablation wave, etc.), the exact solution of the combustion stability problem is of special importance. In this paper we demonstrate the exact analytical solution for the problem of the stability of gasless combustion propagating in solid propellant. An example of such a flame is the combustion of thermites [11]. Since the fuel and the combustion products are in a solid phase, there is no diffusion and the combustion propagates by means of thermal conduction only. It was found [1,2] that combustion in a solid propellant is unstable and the fastest (one dimensional) instability results in a pulsating regime of the combustion front. For the case of large activation energy F. » T2/(T2 —T&), where T~ and T2 are the temperature of the fuel and the combustion products respectively, the chemical reaction zone is thin in comparison with the preheat zone. The solution of the stability problem was obtained in [5—8] by regarding the reaction zone as the inner discontinuity surface and assuming that the temperature perturbation is continuous at the discontinuity surface. The growth rate for the fastest perturbations was found to be