We compute the thermodynamic properties of the glass phase in a binary mixture of soft spheres. Our approach is a generalization to mixtures of the replica strategy, recently proposed by Mezard and Parisi, providing a first principle statistical mechanics computation of the thermodynamics of glasses. The method starts from the interatomic potentials, and translates the problem into the study of a molecular liquid. We compare our analytical predictions to numerical simulations, focusing onto the values of the thermodynamic transition temperature and the configurational entropy.