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Roughly half of the heavy elements (atomic mass greater than that of iron) are believed to be synthesized in the late evolutionary stages of stars with masses between 0.8 and 8 solar masses. Deep inside the star, nuclei (mainly iron) capture neutrons and progressively build up (through the slow-neutron-capture process (1,2), or s-process) heavier elements that are subsequently brought to the stellar surface by convection. Two neutron sources, activated at distinct temperatures, have been proposed: [sup.13]C and [sup.22]Ne, each releasing one neutron per α-particle ([sup.4]He) captured (1-4). To explain the measured stellar abundances (1-7), stellar evolution models invoking the [sup.13]C neutron source (which operates at temperatures of about one hundred million kelvin) are favoured. Isotopic ratios in primitive meteorites, however, reflecting nucleosynthesis in the previous generations of stars that contributed material to the Solar System, point to higher temperatures (more than three hundred million kelvin), requiring at least a late activation of [sup.22]Ne (ref. 1). Here we report a determination of the s-process temperature directly in evolved low-mass giant stars, using zirconium and niobium abundances, independently of stellar evolution models. The derived temperature supports [sup.13]C as the s-process neutron source. The radioactive pair [sup.93]Zr-[sup.93]Nb used to estimate the s-process temperature also provides, together with the pair [sup.99]Tc- [sup.99]Ru, chronometric information on the time elapsed since the start of the s-process, which we determine to be one million to three million years., We obtained high-resolution spectra of 17 S-type stars and 6 M-type stars using the high-resolution (resolving power, λ/Δλ = 85,000) HERMES spectrograph (8) (Extended Data Table 1 and Extended Data [...]