This work carries out a comparative analysis of the stability robustness of PI and PIα controllers when applied to first order plus time delay plants. An analytical result shows that both controllers have exactly the same region of feasible frequency specifications. Nevertheless, the robustness of both controllers is quite different. Depending on the set of frequency specifications and the non integer order of the integral action of the fractional-order controller, the PIα controller can or not provide higher robustness to plant parameters variations than the PI controller. The regions where each controller is most robust are calculated in this paper. Two simulated examples illustrate this opposite robustness behaviour of the PIα controller with respect to the PI controller1.