This paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems. The systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set. Furthermore, these uncertain parameters may be time-varying. The method used to establish asymptotic stability of the closed loop system (obtained when the feedback control is applied) involves the use of a quadratic Lyapunov function. The main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic Lyapunov function. Once the Lyapunov function has been obtained, it is used to construct the stabilizing feedback control law. The fundamental idea behind the algorithm presented involves constructing an upper bound for the Lyapunov derivative corresponding to the closed loop system. This upper bound is a quadratic form. By using this upper bounding procedure, a suitable Lyapunov function can be found by solving a certain matrix Riccati equation. Another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic Lyapunov function.