Coulomb interactions are crucial in determining the ground state of an ideal two-dimensional electron gas (2DEG) in the limit of low electron densities(1). In this regime, Coulomb interactions dominate over single-particle phase-space filling. In silicon and gallium arsenide, electrons are typically localized at these low densities. In contrast, in transition-metal dichalcogenides (TMDs), Coulomb correlations in a 2DEG can be anticipated at experimentally relevant electron densities. Here, we investigate a 2DEG in a gated monolayer of the TMD molybdenum disulfide(2). We measure the optical susceptibility, a probe of the 2DEG which is local, minimally invasive and spin selective(3). In a magnetic field of 9.O T and at electron concentrations up to n similar or equal to 5 x 10(12)cm(-2), we present evidence that the ground state is spin-polarized. Out of the four available conduction bands(4,5), only two are occupied. These two bands have the same spin but different valley quantum numbers. Our results suggest that only two bands are occupied even in the absence of a magnetic field. The spin polarization increases with decreasing 2DEG density, suggesting that Coulomb interactions are a key aspect of the symmetry breaking. We propose that exchange couplings align the spins(6). The Bohr radius is so small(7) that even electrons located far apart in phase-space interact with each other(6).