The phase behaviour of very asymmetric binary mixtures can be understood in terms of the depletion interaction. For hard particles this yields a narrow deep attractive well surrounding the hard core. Colloids with similar interaction potentials are known to destabilize the liquid, causing it to show a wide fluid-solid coexistence, and in extreme cases they exhibit an exotic solid-solid condensation. For a mixture this means that phase separation is not fluid-fluid, as previously thought, but normally fluid-solid, and if the asymmetry is very large, even solid-solid. We present in this work the result of devising a density functional theory for an infinitely asymmetric mixture of parallel hard cubes. This model is singular and undergoes a collapse in a close-packed solid (an extreme fluid-solid demixing). We avoid this collapse by introducing a small amount of polydispersity in the large particles; the resulting phase diagram shows the fluid-solid and solid-solid demixing scenarios described above.