A new molecular simulation procedure is reported for determining the phase behavior of fluids and fluid mixtures, which closely follows the experimental synthetic method. The simulation procedure can be implemented using Monte Calro or molecular dynamics in either the microcanonical or canonical statistical ensembles. Microcanonical molecular dynamics simulations are reported for the phase behavior of both the pure Lennard-Jones fluid and a Lennard-Jones mixture. The vapor pressures for the pure fluid are in good agreement with Monte Carlo Gibbs ensemble and Gibbs-Duhem calculations. The Lennard-Jones mixture is composed of equal size particles, with dissimilar energy parameters (ɛ2/ɛ1 = 1/2, ɛ12/ɛ1 = 1/math). The binary Lennard-Jones mixture exhibits liquid-liquid equilibria at high pressures and the simulation procedure allows us to estimate the coordinates of the high-pressure branch of the critical curve.