A model for the growth of a tumor by diffusion is considered for spheroids which are grown in a normal medium in suspension culture. The experimental evidence indicates that, upon the onset of necrosis in the centre of the spheroid, the viable rim thickness decreases relatively slowly. This evidence contradicts the results of the usual model in which the oxygen consumption per unit volume is assumed to be constant. The proposed model assumes that the oxygen concentration is constant above a critical value, and proportional to the concentration below this critical value. The viable rim thickness of the spheroid, as a function of the necrotic radium, varies markedly over the range of the parameters in this model. By suitably choosing these parameters, the results of this model agree with the experimental evidence.