As part of an extensive multiwavelength monitoring campaign, the International Ultraviolet Explorer satellite was used to observe the broad-line radio galaxy 3C 390.3 during the period 1994 December 31-1996 March 5. Spectra were obtained every 6-10 days. The UV continuum varied by a factor of 7 through the campaign, while the broad emission lines varied by factors of 2-5. Unlike previously monitored Seyfert 1 galaxies, in which the X-ray continuum generally varies with a larger amplitude than the UV, in 3C 390.3 the UV continuum light curve is similar in both amplitude and shape to the X-ray light curve observed by ROSAT. The UV broad emission-line variability lags that of the UV continuum by 35-70 days for Lyalpha and C IV, values larger than those found for Seyfert 1 galaxies of comparable UV luminosity. These lags are also larger than those found for the Balmer lines in 3C 390.3 over the same period. The red and blue wings of C IV and Lyalpha vary in phase, suggesting that radial motion does not dominate the kinematics of the UV line-emitting gas. Comparison with archival data provides evidence for velocity-dependent changes in the Lyalpha and C IV line profiles, indicating evolution in the detailed properties and/or distribution of the broad-line emitting gas.