  Recent experiments in the two dimensional electron systems in silicon MOSFETs
have shown that the in-plane magnetic field $H_{sat}$ required to saturate the
conductivity to its high-field value and the magnetic field $H_s$ needed to
completely align the spins of the electrons are comparable. By small-angle
Shubnikov-de Haas oscillation measurements that allow separate determinations
of the spin-up and spin-down subband populations, we show that $H_{sat}=H_s$ to
an accuracy of 5% for electron densities $n_s > 3 \times 10^{11}$ cm$^{-2}$.
