A monolayer of a transition metal dichalcogenide such as WSe2 is a two-dimensional direct-bandgap valley-semiconductor(1,2) having an effective honeycomb lattice structure with broken inversion symmetry. The inequivalent valleys in the Brillouin zone could be selectively addressed using circularly polarized light fields', suggesting the possibility for magneto-optical measurement and manipulation of the valley pseudospin degree of freedom(6-8). Here we report such experiments that demonstrate the valley Zeeman effect-strongly anisotropic lifting of the degeneracy of the valley pseudospin degree of freedom using an external magnetic field. The valley-splitting measured using the exciton transition deviates appreciably from values calculated using a three-band tight-binding model(9) for an independent electron-hole pair at +/- K valleys. We show, on the other hand, that a theoretical model taking into account the strongly bound nature of the exciton yields an excellent agreement with the experimentally observed splitting. In contrast to the exciton, the trion transition exhibits an unexpectedly large valley Zeeman effect that cannot be understood within the same framework, hinting at a different contribution to the trion magnetic moment. Our results raise the possibility of controlling the valley degree of freedom using magnetic fields in monolayer transition metal dichalcogenides or observing topological states of photons strongly coupled to elementary optical excitations in a microcavity(10).