The mixed H-2/H-infinity control problem can be motivated as a nominal LQG optimal control problem subject to robust stability constraints, expressed in the form of an H-infinity norm bound. While at the present time there exist efficient methods to solve a modified problem consisting on minimizing an upper bound of the H-2 cost subject to the H-infinity constraint, the original problem remains, to a large extent, still open. This paper contains a solution to a general four-block mixed H-2/H-infinity problem, based upon constructing a family of approximating problems. Each one of these problems consists of a finite-dimensional convex optimization and an unconstrained standard H-infinity problem. The set of solutions is such that in the limit the performance of the optimal controller is recovered, allowing one to establish the existence of an optimal solution. Although the optimal controller is not necessarily finite-dimensional, it is shown that a performance arbitrarily close to the optimal can be achieved with rational (and thus physically implementable) controllers. Moreover, the computation of a controller yielding a performance epsilon-away from optimal requires the solution of a single optimization problem, a task that can be accomplished in polynomial time.