This work deals with the problem of rendering the output of a switched linear system insensitive to a disturbance input, with the requirement that the dynamics of the closed-loop system be exponentially stable for all switching signals with a sufficiently large dwell time. A necessary and sufficient condition for the problem to have a solution is shown. A complete computational procedure to check the solvability condition and to synthesize the compensation scheme is provided. A worked-out example is presented with the aim of showing the effectiveness of the devised method.