The spin structure of the valence and conduction bands at the K and K' valleys of single-layer WS2 on Au(111) is determined by spin-and angle-resolved photoemission and inverse photoemission. The bands confining the direct band gap of 1.98 eV are out-of-plane spin polarized with spin-dependent energy splittings of 417 meV in the valence band and 16 meV in the conduction band. The sequence of the spin-split bands is the same in the valence and in the conduction bands and opposite at the K and the K' high-symmetry points. The first observation explains "dark" excitons discussed in optical experiments, the latter points to coupled spin and valley physics in electron transport. The experimentally observed band dispersions are discussed along with band structure calculations for a freestanding single layer and for a single layer on Au(111).

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