This paper is concerned with exploring a convex controller design method for a class of Markovian jump systems with uncertain transition rates. First, a new stability criterion of such systems is established by applying the slack-matrix technique. Then, a totally convex method for the state-feedback controller design is derived in terms of linear matrix inequalities. The proposed design method has the merits of convexity and low conservativeness, while in the previous methods the two merits cannot coexist. Finally, a numerical example is given to illustrate the effectiveness and merits of the proposed method.