We report a computer simulation study of liquid-liquid phase equilibria of symmetrical fluid mixtures. We note that it is not necessary to conduct a conventional Gibbs ensemble simulation-involving two correlated boxes-when studying this type of phase equilibria. We show that by simulation of a single box in the semigrand canonical ensemble under appropriate conditions, one obtains results which are equivalent to those obtained using the conventional Gibbs ensemble method The new method is illustrated with an application to liquid-liquid equilibria of a symmetrical square-well binary mixture. The results of our simulations are compared with two sets of recently reported results for the same system, based on the conventional Gibbs ensemble technique. Finite-size effects are investigated in detail, in order to understand the strong system-size dependence reported for this system, over a wide range of temperatures, using the Gibbs ensemble method at constant volume. A finite-size-scaling analysis is also carried out, to check the consistency of our results in the critical region.