The problem of Rayleigh-Bénard convection at low Prandtl number σ is investigated in a circular geometry. Jones, Moore & Weiss (1976) have formulated, but not solved analytically, an asymptotic nonlinear problem in the limit σ → 0 at small velocities. It is shown that the problem they posed can be solved exactly in this geometry. The solutions are extended by means of expansions in the amplitude ε and the reciprocal of the Reynolds number σε−1, both assumed small. The problem is related to one that occurs in nonlinear mean-field dynamo theory (Malkus & Proctor 1975) and it is surmised that similar problems may be expected to appear in a variety of physical situations.