Multiobjective control problems have been the object of much attention in the past few years, since they allow for handling multiple, perhaps conflicting, performance specifications and model uncertainty. One of the earliest multiobjective problems is the mixed H-2/H-infinity control problem, which can be motivated as a nominal LQG optimal control problem subject to robust stability constraints. This problem has proven to be surprisingly difficult to solve, and at this time no closed-form solutions are available. Moreover, it has been shown that except in some trivial cases, the optimal controller is infinite-dimensional. In this paper, we propose a solution to general continuous-time mixed H-2/H-infinity problems, based upon constructing a family of approximating problems, obtained by solving an equivalent discrete-time problem. Each of these approximations can be solved efficiently, and the resulting controllers converge strongly in the H-2 topology to the optimal solution.