Using the small collector of the Trek experiment, a nuclear track-etch detector deployed on board the Russian space station Mir, we have made a measurement of the isotopic composition of manganese, iron, and nickel in the Galactic cosmic rays (GCR) at energies between 150 and 400 A MeV. We find that the isotopic composition of iron and nickel in the GCR source is consistent with solar system values; in particular, [60Ni]/[58Ni] is 0.78+-0.280.29 times the solar value, in contradiction to two previous results. The measurements were made in 1991-1992, during a maximum in solar activity and unusually strong solar modulation; we estimate that the analyzed GCR ions had kinetic energies between 550 and 1200 A MeV outside of the heliosphere. Based on our upper limit for the abundance of 56Ni, we place a lower limit of ∼24 days on the time between nucleosynthesis and acceleration of GCR material; the lack of significant enhancements in 54Fe or 60Ni excludes the possibility of major contributions of neutron- rich supernova zones to the GCR source. We do not find a strong depletion of 54Mn, indicating that this isotope does not completely decay during propagation. We also place upper limits on the abundances of 54Fe, 57Fe, 58Fe, 62Ni, and 64Ni. Our limit on the abundance of 58Fe is nearly a factor of 3 lower than previously reported upper limits.