  The optical susceptibility is a local, minimally-invasive and spin-selective
probe of the ground state of a two-dimensional electron gas. We apply this
probe to a gated monolayer of MoS$_2$. We demonstrate that the electrons are
spin polarized. Of the four available bands, only two are occupied. These two
bands have the same spin but different valley quantum numbers. We argue that
strong Coulomb interactions are a key aspect of this spontaneous symmetry
breaking. The Bohr radius is so small that even electrons located far apart in
phase space interact, facilitating exchange couplings to align the spins.
