With a new instrument, designed to determine the isotopic composition of Galactic cosmic-ray nuclei, on board the Ulysses spacecraft, we report relative abundance measurements for 54Fe, 55Fe, 56Fe, 57Fe, 58Fe and 58Ni, 59Ni, 60Ni, 62Ni in the overall energy range between ~200 and ~420 MeV nucleon-1. These measurements combine excellent mass resolution (σ ~ 0.28 amu at 56Fe) with good statistical significance (more than 6000 Fe events). The high mass resolution is achieved by utilizing solid-state, position-sensing detectors to measure--with less than 1° angular error--the trajectory of each nucleus entering the charged-particle telescope. The cosmic-ray source abundances are derived from the measurements using models for propagation from distributed sources in the Galaxy (``leaky-box'' model), taking into account solar modulation during penetration of the heliosphere. Overall, except for 54Fe/56Fe and 57Fe/56Fe, we show that the principal Fe and Ni isotopic source ratios have values close to the solar system ratios derived from meteorites. In particular, we note that 58Fe and 62Ni display no evidence of neutron enrichment. We discuss the role of the predominantly electron-capture nuclide 54Mn, which decays mainly by β- in the cosmic rays, and contributes to the abundance of 54Fe measured in the solar system.