We started a series of experiments, aiming at obtaining neutron-capture cross sections from measurements of the production of both long-lived nuclides ¹⁰Be and ¹⁴C, in the energy region of interest for astrophysical applications. Neutron-rich scenarios lead to the production of the long-lived radionuclides ¹⁰Be (t_1/2 = 1.51 Myr) and ¹⁴C (t_1/2 = 5730 yr) via neutron-capture reactions on ⁹Be and ¹³C, respectively. Data for both neutron-capture reactions are sparse and disagree in the astrophysical relevant energy region between 10 and 250 keV. Since (n,γ) cross sections in this mass and energy range are expected to be very small (some tens of µbarn), the combination of the activation technique and accelerator mass spectrometry (AMS) offers the powerful experimental approach which is necessary for such investigations. Beryllium oxide and enriched ¹³C graphite samples have been irradiated in a quasi-stellar neutron spectrum of kT = 25 keV at Forschungszentrum Karlsruhe. The subsequent AMS measurements were performed at the VERA (Vienna Environmental Research Accelerator) laboratory. The first AMS results confirm the expected low cross sections of about 10 µbarn for both reactions.