The magnetic Compton profile of Fe [111] was measured using circularly polarized synchrotron radiation at incident energies of 84.4, 167.2 and 256.0 keV on the high-energy beamline at the European Synchrotron Radiation Facility. It was found that the momentum resolution of these experiments, which use semiconductor detectors, improves by almost a factor of two over what was previously possible by this technique at photon energies of approximately (1/10)mc(2). It was also observed that all three spectra reduced to the magnetic Compton profile, describing the spin-dependent ground-state momentum density, and that within the experimental error the integrated intensity of the magnetic effect scaled as predicted by the cross section derived in the limit of energies much less than the rest energy of the electron. The magnetic Compton profile of Fe [111], measured using 167.2 keV incident energy and with momentum resolution of 0.42 a.u., was compared with the prediction from a full-potential linearized augmented-plane-wave model profile. The fine structure predicted by theory was confirmed by the experimental profile at this improved resolution.