In this article we introduce depletion in asymmetric mixtures within the context of density functional theory. As in the definition of the interaction potentials, it is convenient to work in a semi-grand ensemble at constant chemical potential of the small particles. The free-energy functional can then be mapped onto that of a one-component effective fluid. This method is applied to the fluid of parallel hard cubes, which is studied in the limit of infinite asymmetry. The phase behaviour of this fluid is shown and discussed in terms of the general phase behaviour of additive mixtures of hard particles.