Exchange interaction is responsible for the stability of elementary boson condensates with respect to momentum fragmentation. This remains true for composite bosons when single fermion exchanges are included but spin degrees of freedom are ignored. Here, we show that their inclusion can produce a spin fragmentation of the dark exciton condensate, i.e., an unpolarized condensate with an equal amount of spin (+2) and (-2) excitons not coupled to light. The composite boson many-body formalism allows us to predict that, for spatially indirect excitons, the condensate polarization switches from unpolarized to fully polarized when the distance between the layers confining electrons and holes increases. Importantly, the threshold distance for this switch lies in a regime fully accessible to experiments.