of the RGB, using updated physical inputs and, for the first time, accounting also for atomic diffusion. In view of the peculiar structural properties which characterize Pop. III stars [see @CC93 Paper I], special care was taken to use an appropriate and complete nuclear reaction network. Furthermore, numerical experiments were performed with the aim of testing how the evolution of a metal-free star is affected when its surface is polluted by metal-rich matter through encounters with interstellar gas clouds — a scenario suggested in order to explain the absence of truly zero-metal stars and the relative paucity of extremely metal-poor objects [@Y81]. We found that the metals in the accreted matter fail to reach the nuclear-burning regions, so that the main structural and evolutionary properties of the stars are not affected by the external pollution; it was also clearly shown that it is quite difficult to discriminate observationally between a polluted Pop. III star and a very metal-poor Pop. II object in this phase of evolution.

In Paper I we stopped the computations soon after the onset of the He-flash and did not investigate the evolution during the further development of the He flash at the RGB tip. This had 