In this paper we present the evolution of a low-mass model (initial mass M = 1.5 M⊙) with a very low metal content (Z = 5 × 10−5, equivalent to [Fe/H] = –2.44). We find that, at the beginning of the Asymptotic Giant Branch (AGB) phase, protons are ingested from the envelope in the underlying convective shell generated by the first fully developed thermal pulse. This peculiar phase is followed by a deep third dredge-up episode, which carries to the surface the freshly synthesized 13C, 14N and 7Li. A standard thermally pulsing AGB (TP-AGB) evolution then follows. During the proton-ingestion phase, a very high neutron density is attained and the s process is efficiently activated. We therefore adopt a nuclear network of about 700 isotopes, linked by more than 1200 reactions, and we couple it with the physical evolution of the model. We discuss in detail the evolution of the surface chemical composition, starting from the proton ingestion up to the end of the TP-AGB phase.