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.