As protostars evolve from optically faint/infrared-bright (Class I) sources to optically bright/infrared-faint (Class II) the solid material in their surrounding disks accumulates into planetesimals and protoplanets. The nearby, young Ophiuchus star-forming region contains hundreds of protostars in a range of evolutionary states. Using the Atacama Large Millimeter Array to observe their millimeter continuum emission, we have measured masses of, or placed strong upper limits on, the dust content of 279 disks. The masses follow a log-normal distribution with a clear trend of decreasing mass from less to more evolved protostellar infrared class. The (logarithmic) mean Class I disk mass, M = 3.8 M-circle plus, is about 5 times greater than the mean Class II disk mass, but the dispersion in each class is so high, sigma(logM)similar or equal to 0.8-1, that there is a large overlap between the two distributions. The disk mass distribution of flat-spectrum protostars lies in between Classes I and II. In addition, three Class III sources with little to no infrared excess are detected with low disk masses, M similar or equal to 0.3 M-circle plus. Despite the clear trend of decreasing disk mass with protostellar evolutionary state in this region, a comparison with surveys of Class II disks in other regions shows that masses do not decrease monotonically with age. This suggests that the cloud-scale environment may determine the initial disk mass scale or that there is substantial dust regeneration after 1 Myr.