The production of the bulk of barium has long been ascribed to the main component of the s-process, whose astrophysical site has been envisaged in the convective He shell of Thermally Pulsing Asymptotic Giant Branch (TP-AGB) stars of low mass (1–3 M⊙; see Käppeler et al. 1990). The main neutron source is the 13C(α, n)16O reaction, operating at the thermal energy of kT = 12keV. We have calculated neutron captures in such environment with an updated nuclear physics, adopting the neutron capture cross sections of Beer, Voß, & Winters (1992) together with their temperature-dependence. Stellar models producing a mean neutron exposure of τ0 ≃ 0.30 mb−1 are able to reproduce the solar distribution of the s-abundances satisfactory, but the Ba isotopes show some overproduction. Such a strong indication suggests a revision of the Ba cross sections (see Gallino, Raiteri, & Busso 1992). Once that a suitable choice of σn, γ(Ba) is made, it is found that a r-contribution to solar Ba of the order of 10% can be expected.