Accelerator mass spectrometry (AMS) represents a complementary approach for precise measurements of neutron capture cross sections, e.g., for nuclear astrophysics. This technique, completely independent of previous experimental methods, was applied for the measurement of the Fe54(n,γ)Fe55 reaction. Following a series of irradiations with neutrons from cold and thermal to keV energies, the produced long-lived Fe55 nuclei (t1/2=2.744+-0.009) yr) were analyzed at the Vienna Environmental Research Accelerator. A reproducibility of about 1% could be achieved for the detection of Fe55, yielding cross-section uncertainties of less than 3%. Thus, this method produces new and precise data that can serve as anchor points for time-of-flight experiments. We report significantly improved neutron capture cross sections at thermal energy (σth=2.30±0.07 b) as well as for a quasi-Maxwellian spectrum of kT=25 keV (σ=30.3±1.2 mb) and for En=481±53 keV (σ=6.01±0.23 mb). The new experimental cross sections have been used to deduce improved Maxwellian-averaged cross sections in the temperature regime of the common s-process scenarios. The astrophysical impact is discussed by using stellar models for low-mass asymptotic giant branch stars.