The linear stability of exothermic autocatalytic reaction fronts is considered using the viscous thermohydrodynamic equations for a fluid with finite thermal diffusivity. For upward front propagation and a thin front, the vertical thermal gradient near the front is reminiscent of the Rayleigh-Bénard problem of a fluid layer heated from below. The problem is also similar to flame propagation, except that here the front propagation speed is limited by catalyst diffusion rather than by activation kinetics. For small density changes in a laterally unbounded system, the curvature dependence of the front propagation speed stabilizes perturbations with short wavelengths «<«c, whereas long wavelengths are unstable to convection. The critical wavelengths «c are calculated and compared with experiments and with theoretical results for a similar problem that is driven by a Rayleigh-Taylor instability arising from a discontinuous density difference at the reaction front. © 1992 The American Physical Society.