The linear polarization degree of the photoluminescence from a nonresonantly pumped CdTe microcavity exhibits a dramatic increase from about 0 to more than 85% once the threshold of stimulated scattering of the exciton polaritons toward the Bose-Einstein condensate is passed. The linear polarization direction is pinned to one of the crystallographic axes of the cavity. These effects are well accounted for within a model of localized condensate of interacting bosons.