  We analyze Hubble Space Telescope observations of six globular clusters in
the Large Magellanic Cloud from program GO-14164 in Cycle 23. These are the
deepest available observations of the LMC globular cluster population; their
uniformity facilitates a precise comparison with globular clusters in the Milky
Way. Measuring the magnitude of the main sequence turnoff point relative to
template Galactic globular clusters allows the relative ages of the clusters to
be determined with a mean precision of 8.4%, and down to 6% for individual
objects. We find that the mean age of our LMC cluster ensemble is identical to
the mean age of the oldest metal-poor clusters in the Milky Way halo to 0.2
$\pm$ 0.4 Gyr. This provides the most sensitive test to date of the
synchronicity of the earliest epoch of globular cluster formation in two
independent galaxies. Horizontal branch magnitudes and subdwarf fitting to the
main sequence allow us to determine distance estimates for each cluster, and
examine their geometric distribution in the LMC. Using two different methods,
we find an average distance to the LMC of 18.52 $\pm$ 0.05.
