The Am241(n,γ) cross section has been measured at the n-TOF facility at CERN with the n-TOF BaF2 Total Absorption Calorimeter in the energy range between 0.2 eV and 10 keV. Our results are analyzed as resolved resonances up to 700 eV, allowing a more detailed description of the cross section than in the current evaluations, which contain resolved resonances only up to 150-160 eV. The cross section in the unresolved resonance region is perfectly consistent with the predictions based on the average resonance parameters deduced from the resolved resonances, thus obtaining a consistent description of the cross section in the full neutron energy range under study. Below 20 eV, our results are in reasonable agreement with JEFF-3.2 as well as with the most recent direct measurements of the resonance integral, and differ up to 20-30% with other experimental data. Between 20 eV and 1 keV, the disagreement with other experimental data and evaluations gradually decreases, in general, with the neutron energy. Above 1 keV, we find compatible results with previously existing values.