A mixed sensitivity H-infinity problem is solved for dead-time systems. It is shown that for a given bound on the H-infinity-norm causal stabilizing controllers exist that achieve this bound if and only if a related finite-dimensional Riccati equation has a solution with a certain nonsingularity property. In the case of zero time delay, the Riccati equation is a standard Riccati equation and the nonsingularity condition is that the solution be nonnegative definite, For nonzero time delay, the nonsingularity condition is more involved but still allows us to obtain controllers. All suboptimal controllers are parameterized, and the central controller is shown to be a feedback interconnection of a finite-dimensional system and a finite memory system, both of which can be implemented, Some H-infinity problems are rewritten as pure rational H-infinity problems using a Smith predictor parameterization of the controller.