Most physical processes exhibit transport delay in the measured output, and it is well known that this can have disastrous effects on system stability and performance if it is not accounted for. In this paper, we give necessary and sufficient conditions for existence of estimators and controllers that achieve the desired H-infinity, performance criterion when such a measurement delay is present. We also give the complete characterization of all controllers and estimators that achieve the desired performance criterion. The necessary and sufficient conditions are easy to check and are given in terms of the familiar pair of algebraic Riccati equations that appear in the nondelay versions of the corresponding H-infinity problems, along with an additional Riccati differential equation. Explicit state-space formulas for the controllers and estimators are also obtained. They have a linear periodic structure and are easily implementable. To obtain these results, we first obtain state-space results for a "modified" Nehari problem, which may be of independent interest (see Problem 5 in section 2).