In relativistic heavy-ion collisions, the strong Lorentz-contracted electromagnetic fields are capable of producing copious numbers of lepton pairs through the two-photon mechanism. We provide a new and more general derivation of the Poisson distribution for multiple-pair production and develop a hybrid Monte Carlo technique for calculating the impact parameter dependence and the related multiple-pair production in the two-photon external-field model. Computations for 200 GeV/nucleon sulfur incident on a gold target are compared with recent data. There is good agreement for all but the most energetic pairs, where the theory underpredicts the data. We investigate the ability of the equivalent-photon approximation to predict the impact-parameter dependence of the process, as well as multipair production. We find that the approximation is limited in its ability to address these phenomena. (C) 1999 Academic Press.