The problem of stabilization of a linear system characterized by its rational transfer function, with an input time delay, by a rational controller, is considered. If the delay is not known at all (its interval is infinite), the necessary and sufficient condition for the existence of a rational controller is stability of the open-loop rational giant. Moreover, the existence of a rational stabilizing controller implies the existence of a constant gain stabilizing controller. If the delay is known to lie in a given finite interval, two non-existence theorems are derived, one for the constant gain stabilizing controller and one for any rational stabilizing controller. A design method for stabilizing rational controllers, based on first-order all-pass filters cascaded by a constant gain and on violating the conditions for the non-existence theorem of the constant gain stabilizer, is presented.