One of the main neutron sources for the astrophysical s process is the reaction 13C(α,n)16O, taking place in thermally pulsing asymptotic giant branch stars at temperatures around 90 MK. To model the nucleosynthesis during this process the reaction cross section needs to be known in the 150–230 keV energy window (Gamow peak). At these sub-Coulomb energies, cross section direct measurements are severely affected by the low event rate, making us rely on input from indirect methods and extrapolations from higher-energy direct data. This leads to an uncertainty in the cross section at the relevant energies too high to reliably constrain the nuclear physics input to s-process calculations. We present the results from a new deep-underground measurement of 13C(α,n)16O, covering the energy range 230–300 keV, with drastically reduced uncertainties over previous measurements and for the first time providing data directly inside the s-process Gamow peak. Selected stellar models have been computed to estimate the impact of our revised reaction rate. For stars of nearly solar composition, we find sizeable variations of some isotopes, whose production is influenced by the activation of close-by branching points that are sensitive to the neutron density, in particular, the two radioactive nuclei 60Fe and 205Pb, as well as 152Gd.