An observer-based approach to perfect disturbance rejection in linear continuous systems with delayed control is presented and exemplified by an application to a peristaltic pump. Disturbance rejection properties of the Smith Predictor are studied. A recently developed technique of continuous deadbeat observation is exploited to predict the disturbance signal which is described by a known dynamic model. It is shown that under certain algebraic conditions a disturbance estimate based control law is able to provide a disturbance-free output signal. The disturbance rejection scheme can be used in combination with any conventional control technique such as stabilizing or servo controller.