A general and rigorous method for the stability analysis of a spinning body which is part rigid and part elastic is presented. The motion of the body is described by a system of equations, namely a system consisting of both ordinary and partial differential equations. The stability analysis is based on the Liapunov direct method and it works directly with the hybrid set of equations in contrast with the common practice of replacing the partial differential equations with ordinary differential equations by means of modal truncation or spatial discretization. This new approach permits a more rigorous analysis which not only produces sharper stability criteria but also avoids questions as to the effect of the truncation and discretization processes on the results. The general formulation can be used to test the stability of orbiting satellites with various types of elastic members and should also be applicable to the stability analysis of hybrid systems encountered in other areas.