Electronic structure of antiferromagnetic zigzag-graphene-nanoribbon/graphene is investigated by performing first-principles and tight-binding calculations. We show that the unique combination of spin and pseudospin in ZGNRs can be used to manipulate properties of graphene due to their perfect lattice match. In the Bernal stacking of the two constituents, although they are interacting via van der Waals force, half-metallicity is surprisingly obtained due to spin- and pseudospin-symmetry breaking. In particular, half-metallic Dirac cone is formed at K (K') near the Fermi level. The present finding not only is of fundamental interest but also has practical implications in potential applications of graphene-based nanoelectronics.