Polarized neutron scattering has been used to determine the changes in the distribution of unpaired electrons which take place in the martensitic transition in Ni₂MnGa. Ni₂MnGa is a ferromagnetic Heusler alloy which undergoes a reversible transition at about 220 K from a high temperature cubic phase to a low temperature tetragonal one. It has been suggested, on the basis of band structure calculations, that the structural phase transition is driven by a band Jahn-Teller distortion involving redistribution of electrons between 3d sub-bands of different symmetries. The results of the neutron scattering experiments show that the transition from the cubic to the tetragonal phase is accompanied by a transfer of magnetic moment from Mn to Ni. The unpaired electrons in the cubic phase have overall e_g symmetry. In the tetragonal phase, the degeneracy of the e_g and t_2g bands is raised and the unpaired electrons are redistributed in such a way that the sub-bands based on orbitals extending towards the c-axis are preferentially occupied. Although the experimental moments differ in detail from those expected from band structure calculations, the change in symmetry of the magnetization distribution is consistent with a band Jahn-Teller origin for the phase transition.