  The large and diffuse galaxies NGC1052-DF2 and NGC1052-DF4 have been found to
have very low dark matter content and a population of luminous globular
clusters. Accurate distance measurements are key to interpreting these
observations. Recently, the distance to NGC1052-DF4 was found to be $20.0\pm
1.6$ Mpc by identifying the tip of the red giant branch (TRGB) in 12 orbits of
Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS) imaging. Here we
present 40 orbits of HST ACS data for NGC1052-DF2 and use these data to measure
its TRGB. The TRGB is readily apparent in the color-magnitude diagram. Using a
forward model that incorporates photometric uncertainties, we find a TRGB
magnitude of $m_{\rm F814W, TRGB} = 27.67 \pm 0.10$ mag. The inferred distance
is $D_{\rm TRGB} = 22.1 \pm 1.2$ Mpc, consistent with the previous surface
brightness fluctuation distances to the bright elliptical galaxy NGC1052. The
new HST distance rules out the idea that some of NGC1052-DF2's unusual
properties can be explained if it were at $\sim 13$ Mpc; instead, it implies
that the galaxy's globular clusters are even more luminous than had been
derived using the previous distance of 20 Mpc. The distance from NGC1052-DF2 to
NGC1052-DF4 is well-determined at $2.1\pm 0.5$ Mpc, significantly larger than
the virial diameter of NGC1052. We discuss the implications for formation
scenarios of the galaxies and for the external field effect, which has been
invoked to explain the intrinsic dynamics of these objects in the context of
modified Newtonian dynamics.
