We report a measurement of the nuclear polarization of laser-cooled, optically pumped 37K atoms which will allow us to precisely measure angular correlation parameters in the ${\beta }^{+}$-decay of the same atoms. These results will be used to test the V ₋ A framework of the weak interaction at high precision. At the Triumf neutral atom trap (Trinat), a magneto-optical trap confines and cools neutral 37K atoms and optical pumping spin-polarizes them. We monitor the nuclear polarization of the same atoms that are decaying in situ by photoionizing a small fraction of the partially polarized atoms and then use the standard optical Bloch equations to model their population distribution. We obtain an average nuclear polarization of $\bar{P}=0.9913\pm 0.0009$, which is significantly more precise than previous measurements with this technique. Since our current measurement of the β-asymmetry has $0.2 \% $ statistical uncertainty, the polarization measurement reported here will not limit its overall uncertainty. This result also demonstrates the capability to measure the polarization to $\lt 0.1 \% $, allowing for a measurement of angular correlation parameters to this level of precision, which would be competitive in searches for new physics.