  We investigate the evolution of initially metal-free, low-mass Red Giant
stars through the He core flash at the tip of the Red Giant Branch. The low
entropy barrier between the helium- and hydrogen-rich layers enables a
penetration of the helium flash driven convective zone into the inner tail of
the extinguishing H-burning shell. As a consequence, protons are mixed into
high-temperature regions triggering a H-burning runaway. The subsequent
dredge-up of matter processed by He and H burning enriches the stellar surface
with large amounts of helium, carbon and nitrogen. Extending previous results
by Hollowell et al. (1990) and Fujimoto et al. (2000), who claimed that the
H-burning runaway is an intrinsic property of extremely metal-poor low-mass
stars, we found that its occurrence depends on additional parameters like the
initial composition and the treatment of various physical processes.
  We perform some comparisons between predicted surface chemical abundances and
observational measurements for extremely metal-deficient stars. As in previous
investigations, our results disclose that although the described scenario
provides a good qualitative agreement with observations, considerable
discrepancies still remain. They may be due to a more complex evolutionary path
of `real' stars, and/or some shortcomings in current evolutionary models.
  In addition, we analyze the evolutionary properties after the He core flash,
during both the central and shell He-burning phases, allowing us to deduce some
interesting differences between models whose Red Giant Branch progenitor has
experienced the H-flash and canonical models. In particular, the Asymptotic
Giant Branch evolution of extremely metal-deficient stars and the occurrence of
thermal pulses are strongly affected by the previous RGB evolution.
