We consider a mixed H-infinity/H-2 control problem, in which a controller is required to achieve the best possible rejection of a stochastic disturbance with given statistics, subject to an H-infinity-performance constraint. We assume that the H-infinity-constraint rules out the use of the H-2-optimal (LQG) control law. As a result, the H-2-performance of the controller must be degraded. Moreover, no control law can be the best with respect to all the possible measurement signals. It is thus reasonable to look for a controller that will achieve the best H-2-performance, subject to the worst possible measurement signal. An optimal controller, in the sense of this paper, minimizes the worst-case ratio between the resulting H-2-performance degradation and the performance degradation that would result if one had used a standard H-infinity-central controller. In this setting, the mixed H-infinity/H-2 problem reduces to a standard two-person, zero-sum, linear quadratic dynamic game, with a state-feedback information pattern. Both the continuous and the discrete-time cases are considered.