We report a numerical study of homogeneous gas-liquid nucleation in a binary mixture. We study the size and the composition of the critical nucleus as a function of the composition and supersaturation of the vapor. As we make the (Lennard-Jones) mixture increasingly nonideal, we find that there is a regime where the critical nucleus is still miscible in all proportions, even though the bulk liquid phase is not. When these critical nuclei grow, their composition "bifurcates" to approach the value of one of the two bulk phases. For more strongly nonideal mixtures, the two species in the critical nucleus are no longer completely miscible:we observe droplets that are either rich in one species, or in the other. However, we do not find evidence for phase separation inside the critical nucleus-a scenario suggested by Talanquer and Oxtoby [J. Chem. Phys. 104, 1993 (1996)]. In fact, our simulations show that such demixed clusters have a higher free energy than critical nuclei that have an asymmetric composition.