Seven precise eclipse times for the cataclysmic variable star U Geminorum have been obtained during the 1974-1975 observing season. These, together with 65 others, have been used to reevaluate the eclipse parameters. With some selection of the eclipses used, a statistically significant slowdown of the binary system is found. Using previously published masses for the components of U Gem, a mass transfer rate of 81 billionths of a solar mass per year away from the primary is inferred. This direction of mass transfer is opposite to that required by current theoretical understanding of these systems.