We report the discovery of doubly deuterated water (D2O, heavy water) in the interstellar medium. Using the James Clerk Maxwell Telescope and the Caltech Submillimeter Observatory 10 m telescope, we detected the 110-101 transition of para-D2O at 316.7998 GHz in both absorption and emission toward the protostellar binary system IRAS 16293-2422. Assuming that the D2O exists primarily in the warm regions where water ices have been evaporated (i.e., in a ``hot corino'' environment), we determine a total column density of N(D2O) of 1.0×1013 cm-2 and a fractional abundance of D2O/H2=1.7×10-10. The derived column density ratios for IRAS 16293-2422 are D2O/HDO=1.7×10-3 and D2O/H2O=5×10-5 for the hot corino gas. Steady state models of water ice formation, either in the gas phase or on grains, predict D2O/HDO ratios that are about 4 times larger than that derived from our observations. For water formation on grain surfaces to be a viable explanation, a larger H2O abundance than that measured in IRAS 16293-2422 is required. Alternatively, the observed D2O/HDO ratio could be indicative of gas-phase water chemistry prior to a chemical steady state being attained, such as would have occurred during the formation of this source. Future observations with the Herschel Space Observatory satellite will be important for settling this issue.