Electronic structures of ordered double perovskites $Ba_2MReO_6$ (M=Mn, Fe, Co, and Ni) are investigated by using the linearized muffin-tin orbitals band method in the local spin-density approximation (LSDA) and the LSDA+U method. The half-metallic ferrimagnetic ground states are obtained for M=Fe and Ni in the LSDA+U, whereas the insulating ground state is obtained for M=Mn in the LSDA+U incorporating the spinorbit interaction. For M= Co, the antiferromagnetic ground state is stabilized in the LSDA+U by invoking the structural distortion.