The ratio of the neutron-induced fission cross-sections of ²⁴³Am and ²³⁵U was measured in the energy range from 0.5 to 20 MeV with uncertainties of ≈ 4%. The experiment was performed at the CERN n_TOF facility using a fast ionization chamber. With the good counting statistics that could be achieved thanks to the high instantaneous flux and the low backgrounds, the present results are useful for resolving discrepancies in previous data sets and are important for future reactors with improved fuel burn-up.