The energy spectrum of fast neutrons generated by interactions between galactic cosmic rays and the Mars Odyssey spacecraft during quiet interplanetary conditions is determined. It is given by Φn(En) = (0.071 ¿ 0.0044) En-(0.28¿0.055) cm-2 s-1 MeV-1 for energies between 0.7 and 6 MeV. The estimated ratio of this component of spacecraft neutron background to the Martian leakage flux in Mars orbit is expected to amount to between 0.30 and 0.34. This fraction is sufficiently small so that the total background in orbit about Mars is expected to be dominated by Martian leakage neutrons reprocessed by the spacecraft. Nevertheless, the measured asymmetry properties of the Odyssey Neutron Spectrometer response function should allow separation of foreground and background neutron fluxes when in Mars orbit. The fast neutron flux between 0.7 and 6 MeV should only contribute 0.40 Roentgen Equivalent Man (rem) per year to the total radiation dose aboard a Mars Odyssey class spacecraft at solar maximum.