Convective (or spatial) instability, which manifests itself as the tuned amplification of perturbations in the course of their propagation along a non-isothermal packed-bed tubular reactor, is shown to occur in the exothermic standard reaction A --> B + heat. The instability is caused by the interplay of the differential transport of heat and matter and of the activator-inhibitor kinetics inherent in non-isothermal, exothermic reactions (where heat plays the role of autocatalytic species or activator, and matter represents the inhibitor). The differential transport is caused by the inert reactor packing which acts as a thermal reservoir and slows down the diffusive and advective transport of heat relative to that of matter. This instability appears to be relevant to an earlier observation (Puszynski and Hlavacek, 1980, Chem. Engng Sci. 35, 1769-1774) of sustained temperature oscillations in a packed-bed reactor at high Lewis number.