The emission of dileptons from a matter distribution undergoing longitudinal hydrodynamic expansion as might be produced in ultrarelativistic nuclear collisions is computed. Contributions come from quark-gluon plasma, from pion gas, and from a mixed phase while the matter undergoes a first-order confinement phase transition. This yield is compared with that expected from other sources. It is found that the signals from quark-gluon plasma and pion gas probably exceed such background for sufficiently high initial temperature. The ratio of dilepton rapidity density to the square of the pion rapidity density may provide a measurement of the phase transition temperature. The dilepton mass distribution for various values of transverse energy may provide a measure of vector-meson masses as a function of temperature, and a mass shift due to the restoration of chiral symmetry might be seen.