  We present Period-Luminosity Relations (PLRs) for 55 Cepheids in M31 with
periods ranging from 4 to 78 days observed with the Hubble Space Telescope
(HST) using the same three-band photometric system recently used to calibrate
their luminosities. Images were taken with the Wide Field Camera 3 in two
optical filters (F555W and F814W) and one near-infrared filter (F160W) using
the Drift And SHift (DASH) mode of operation to significantly reduce overheads
and observe widely-separated Cepheids in a single orbit. We include additional
F160W epochs for each Cepheid from the Panchromatic Hubble Andromeda Treasury
(PHAT) and use light curves from the Panoramic Survey Telescope and Rapid
Response System of the Andromeda galaxy (PAndromeda) project to determine mean
magnitudes. Combined with a 1.28$\%$ absolute calibration of Cepheid PLRs in
the Large Magellanic Cloud from Riess et al. (2019a) in the same three filters,
we find a distance modulus to M31 of $\mu_0$ = 24.407 $\pm$ 0.032,
corresponding to 761 $\pm$ 11 kpc and 1.49$\%$ uncertainty including all error
sources, the most precise determination of its distance to date. We compare our
results to past measurements using Cepheids and the Tip of the Red Giant Branch
(TRGB). This study also provides the groundwork for turning M31 into a
precision anchor galaxy in the cosmic distance ladder to measure the Hubble
constant together with efforts to measure a fully geometric distance to M31.
