This work considers the stabilization of a high-order system with time delay; an observer–predictor scheme is designed to estimate an internal signal of the system that is not available for measurement: this internal signal is the output before being delayed. By using the estimated signal, it is possible to design a controller for the delay-free system. The key point to carrying out this estimation strategy is to obtain conditions assuring that the estimated signal converges to the internal variable of the system. A necessary and sufficient condition to achieve an appropriate convergence in the proposed observer–predictor scheme is given. In addition, an analysis of the disturbance rejection and robustness with respect to the delay term is provided. The correct functioning of this scheme is verified through an example.