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The Zeeman splittings and EPR frequencies of alkali-metal atoms are shifted in the presence of a polarized noble gas. For a spherical geometry, the shift is enhanced over what is expected classically by a dimensionless atomic parameter $\kappa_0$ that is unique to each alkali-metal atom - noble-gas pair. We present a precise measurement of $\kappa_0$ for the $^{39}$K-$^3$He system with a relative accuracy of better than 1\%. A critical component of achieving sub-percent accuracy involved characterizing the shape of our samples using both MRI and CT medical-imaging techniques. The parameter $\kappa_0$ plays an important role in establishing the absolute polarization of $^3$He in a variety of contexts, including polarized targets for electron scattering experiments and MRI of the gas space of the lungs. Our measurement more than doubles the accuracy possible when using $\kappa_0$ for polarimetry purposes. Just as important, the work presented here represents the first {\it direct} measurement of $\kappa_0$ for the $^{39}$K-$^3$He system; previous values for $\kappa_0$ in the $^{39}$K-$^3$He system relied on a chain of measurements that were benchmarked by previous measurements of $\kappa_0$ in the Rb-$^3$He system., Comment: 18 pages, 15 figures