We consider the process of chemical erosion of a porous medium infiltrated by a reactive fluid in a thin-front limit, in which the width of the reactive front is negligible with respect to the diffusive length. We show that in the radial geometry the advancing front becomes unstable only if the flow rate in the system is sufficiently high. The existence of such a stable region in parameter space is in contrast to the Saffman-Taylor instability in radial geometry, where for a given flow rate the front always eventually becomes unstable, after reaching a certain critical radius. We also examine the similarities between the reactive-infiltration instability and the similar instability in the heat transfer, which is driving the formation of star-like patterns on frozen lakes.