  New stellar models with mass ranging between 4 and 8 Mo, Z=0 and Y=0.23 are
presented. The models have been evolved from the pre Main Sequence up to the
Asymptotic Giant Branch (AGB). At variance with previous claims, we find that
these updated stellar models do experience thermal pulses in the AGB phase. In
particular we show that: a) in models with mass larger than 6 Mo, the second
dredge up is able to raise the CNO abundance in the envelope enough to allow a
"normal" AGB evolution, in the sense that the thermal pulses and the third
dredge up settle on; b) in models of lower mass, the efficiency of the CNO
cycle in the H-burning shell is controlled by the carbon produced locally via
the 3alpha reactions. Nevertheless the He-burning shell becomes thermally
unstable after the early AGB. The expansion of the overlying layers induced by
these weak He-shell flashes is not sufficient by itself to allow a deep
penetration of the convective envelope. However, immediately after that, the
maximum luminosity of the He flash is attained and a convective shell
systematically forms at the base of the H-rich envelope. The innermost part of
this convective shell probably overlaps the underlying C-rich region left by
the inter-shell convection during the thermal pulse, so that fresh carbon is
dredged up in a "hot" H-rich environment and a H flash occurs. This flash
favours the expansion of the outermost layers already started by the weak
thermal pulse and a deeper penetration of the convective envelope takes place.
Then, the carbon abundance in the envelope rises to a level high enough that
the further evolution of these models closely resembles that of more metal rich
AGB stars. These stars provide an important source of primary carbon and
nitrogen.
