The classical equation for electron thermal conductivity, which gives a heat flux proportional to the temperature gradient, breaks down in regions of the Venus ionosphere where the temperature scale height becomes comparable to or less than the electron mean free path. Assuming that the actual flux is approximaely given by the harmonic mean of the classical and the free molecular flow heat flux, we calculate model temperature profiles based on the Pioneer Venus data. We find that the high dayside and nightside temperatrures can be maintained, in the absence of a magnetic field, with somewhat lower heat inputs than those suggested by previous investigators who also neglected the magnetic field. The required heat influx on the nightside is only about one fourth that on the dayside, in sharp contrast to the results of earlier calculations.