Monolayer transition metal dichalcogenide crystals, as direct-gap materials with strong light-matter interactions, have attracted much recent attention. Because of their spin-polarized valence bands and a predicted spin splitting at the conduction band edges, the lowest-lying excitons in WX2 (X = S, Se) are expected to be spin-forbidden and optically dark. To date, however, there has been no direct experimental probe of these dark excitons. Here, we show how an in-plane magnetic field can brighten the dark excitons in monolayer WSe2 and permit their properties to be observed experimentally. Precise energy levels for both the neutral and charged dark excitons are obtained and compared with ab initio calculations using the GW-BSE approach. As a result of their spin configuration, the brightened dark excitons exhibit much-increased emission and valley lifetimes. These studies directly probe the excitonic spin manifold and reveal the fine spin-splitting at the conduction band edges.