The technique of accelerator mass spectrometry (AMS) offers a complementary tool for studying long-lived radionuclides in nuclear astrophysics: (1) as a tool for investigating nucleosynthesis in the laboratory; and (2) via a direct search of live long-lived radionuclides in terrestrial archives as signatures of recent nearby supernova-events. A key ingredient to our understanding of nucleosynthesis is accurate cross-section data. AMS was applied for measurements of the neutron-induced cross sections 13C(n,γ) and 14N(n,p), both leading to the long-lived radionuclide 14C. Solid samples were irradiated at Karlsruhe Institute of Technology with neutrons closely resembling a Maxwell–Boltzmann distribution for kT = 25 keV, and with neutrons of energies between 123 and 178 keV. After neutron activation the amount of 14C nuclides in the samples was measured by AMS at the VERA (Vienna Environmental Research Accelerator) facility. Both reactions, 13C(n,γ)14C and 14N(n,p)14C, act as neutron poisons in s-process nucleosynthesis. However, previous experimental data are discordant. The new data for both reactions tend to be slightly lower than previous measurements for the 25 keV Maxwell–Boltzmann energy distribution. For the higher neutron energies no previous data did exist for 13C(n,γ), but model calculations indicated a strong resonance structure between 100 and 300 keV which is confirmed by our results. Very limited information is available for 14N(n,p) at these energies. Our new data at 123 and 178 keV suggest lower cross sections than expected from previous experiments and data evaluations.