  The spin structure of the valence and conduction bands at the
$\overline{\text{K}}$ and $\overline{\text{K}}$' valleys of single-layer WS$_2$
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 $\overline{\text{K}}$ and the $\overline{\text{K}}$'
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).
