The technique of accelerator mass spectrometry (AMS), offering a complementary tool for sensitive studies of key reactions in nuclear astrophysics, was applied for measurements of the 13 C(n,γ ) 14 C and the 14 N(n,p) 14 C cross sections, which act as a neutron poison in s-process nucleosynthesis. Solid samples were irradiated at Karlsruhe Institute of Technology with neutrons closely resembling a Maxwell-Boltzmann distribution for kT = 25 keV, and also at higher energies between E n = 123 and 182 keV. After neutron irradiation the produced amount of 14 C in the samples was measured by AMS at the Vienna Environmental Research Accelerator (VERA) facility. For both reactions the present results provide important improvements compared to previous experimental data, which were strongly discordant in the astrophysically relevant energy range and missing for the comparably strong resonances above 100 keV. For 13 C(n,γ ) we find a four times smaller cross section around kT = 25 keV than a previous measurement. For 14 N(n,p), the present data suggest two times lower cross sections between 100 and 200 keV than had been obtained in previous experiments and data evaluations. The effect of the new stellar cross sections on the s process in low-mass asymptotic giant branch stars was studied for stellar models of 2 M initial mass, and solar and 1/10 th solar metallicity.