  A monolayer of a transition metal dichalcogenide (TMD) such as WSe$_2$ is a
two-dimensional (2D) direct band-gap valley-semiconductor 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. 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. While the
valley-splitting measured using the exciton transition is consistent with the
difference of the conduction and valence band orbital magnetic moments, the
trion transition exhibits an unexpectedly large valley Zeeman effect which
cannot be understood using an independent electron-hole picture. Instead, we
find an explanation using the recently predicted large Berry curvature and the
associated magnetic moment for the electron-hole exchange interaction modified
trion dispersion. Our results raise the possibility of observing optical
excitation induced valley Hall effect in monolayer TMDs or topological states
of photons strongly coupled to trion excitations in a microcavity.
