The model penetrative-convection problem of ice-water convection is considered. Analytical progress is made through the remarkable simplification that horizontally long convection cells are preferred when the heat flux is fixed on the boundaries [C. J. Chapman and M. R. E. Proctor, ibid. 101, 759-782 (1980; Zbl 0507.76049)]. However, a linear analysis shows that long horizontal scales are preferred only when the convection is mildly penetrative (i.e. the overlying layer of stable fluid is not deep). A straightforward nonlinear asymptotic analysis of the convection only provides the relatively uninteresting information that the convection is subcritical. Using the technique of reconstitution [the author, SIAM J. Math. Anal. 16, 1243-1258 (1985; Zbl 0582.76060)] to provide higher-order corrections to the asymptotic theory, flow properties at larger amplitudes are calculated and predictions about the extent of the subcriticality are made.