344 authors from 52 institutions, 7 pages, 4 figures. Submitted to Physical Review Letters. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htmlThe PHENIX experiment has measured the dielectron continuum in sqrt{s_NN} = 200 GeV Au+Au collisions. In minimum bias collisions the dielectron yield in the mass range between 150 and 750 MeV/c^2 is enhanced by a factor of 3.4 +/- 0.2(stat.) +/- 1.3(syst.) +/- 0.7(model) compared to the expectation from our model of hadron decays. The integrated yield increases faster with the centrality of the collisions than the number of participating nucleons, suggesting emission from scattering processes in the hot and dense medium. The continuum yield between the masses of the phi and the J/psi mesons is consistent with expectations from correlated c-cbar production, though other mechanisms are not ruled out