In this paper we present a Bounded Real Lemma for continuous-time Markov jump linear systems with partial observations of the Markov chain. The main result provides a fundamental tool for the development of an H∞ control theory devoted to this class of systems, and consists of a necessary and sufficient condition for a given stochastic system to be mean square stable and satisfy a prescribed degree of H∞ performance. We also devise a method for the design of H∞ controllers via linear matrix inequalities, which is subsequently applied to the control of an unmanned aerial vehicle model.