This paper proposes the design of a stabilizing output-feedback linear quadratic controller for a pure input-delayed system. By reducing a pure input-delayed system into a non-delayed system and modifying the linear quadratic cost, a simple, stabilizing output-feedback linear quadratic controller is formulated in terms of linear matrix inequalities. The worst-case output-feedback LQ controller is also presented with respect to the unknown initial states. A numerical example is given to illustrate the performance of the proposed controllers.