The constraint imposed by rapid rotation stabilizes convective motions in a plane liquid layer, so that the critical Rayleigh number which marks their onset increases as the two-thirds power of the Taylor number. Here we formulate an asymptotic approximation to the equations governing an analogous problem in a sphere. The solutions we obtain for the asymptotic equations predict the same power law for the dependence of the corresponding critical Rayleigh number on the Taylor number. The analysis is limited at the outset to the case of axially symmetric motions, and the present result, taken in conjunction with earlier results for slow rotation, essentially completes the treatment of symmetric, convective modes of instability.