This paper presents a Galerkin method to solve the mixed H-2/H-infinity optimal control problem. In this method the generator set is built step-by-step by an optimization procedure. Algorithm convergence is proved in the setting of weighted Hardy spaces. Some examples are discussed, showing the accelerated convergence obtained by this new algorithm and its numerical features. A particular quadratic optimal control problem on finite-dimensional linear systems under robustness constraints is developed as a motivation for the mixed H-2/H-infinity problem.