We have obtained spectroscopy with the Far Ultraviolet Spectroscopic Explorer (FUSE) of the supersoft X-ray binary RX J0513.9-6951 over a complete binary orbital cycle. The spectra show a hot continuum with extremely broad O VI emission and weak Lyman absorptions. He II emission is weak and narrow, while N III and C III emissions are undetected, although lines from these ions are prominent at optical wavelengths. The broad O VI emission and Lyman absorption show radial velocity curves that are approximately antiphased and have semiamplitudes of ~117 +- 40 and 54 +- 10 km/s, respectively. Narrow emissions from He II and O VI show small velocity variations with phasing different from the broad O VI, but consistent with the optical line peaks. We also measure considerable changes in the FUV continuum and O VI emission line flux. We discuss the possible causes of the measured variations and a tentative binary interpretation.