There exists a disturbing controversy in the literature about the sign of the Soret effect in binary mixtures of model fluids (Lennard-Jones atoms), whose components differ only in their molecular diameter. For such mixtures, the dependence of the Soret coefficient on the state (liquid versus supercritical), on the system size and on details of handling the range and the cutoff of the Lennard-Jones potential is examined by molecular-dynamics simulations. We establish unambiguously the direction of the Soret effect: Under all circumstances investigated, large particles are driven to the hot region. At supercritical densities, the Soret effect is considerably smaller than in the dense liquid and, furthermore, details of the attractive tail of the Lennard-Jones potential become much more important.