The production of prompt electron-positron pairs in 16 GeV/c .pi.−p collisions has been measured using the LASS spectrometer at SLAC. An excess of events is observed above the estimated contributions of direct and Dalitz decay of known resonances in the kinematic range defined by 0.1 less than or equal to x less than or equal to 0.45, 0 less than or equal to P/sub T/ less than or equal to 0.8 GeV/c and 0.2 less than or equal to M(ee−) less than or equal to 0.7 GeV/c2. The excess signal decreases slowly with increasing M, but exhibits very steep x and P/sub T/2 dependence. The contribution of this signal to the e+e−/.pi./sup +/.pi.− and .gamma./.pi. ratios is discussed. Detailed comparisons are made between e/sup +/e− distributions and the corresponding low mass .mu./sup +/.mu.− distributions, and a simple production mechanism is proposed which describes the 16 GeV/c data well. The implications for direct photon production are presented, and it is shown that the model provides simultaneously a good description of the experimental data on the (e/.pi.) and (.mu./.pi.) ratios for p/sub T/