  We present a low-dispersion period-luminosity relation (PL) based on 154
Cepheids in Messier 33 (M33) with Hubble Space Telescope (HST) photometry from
the PHATTER survey. Using high-quality ground-based light curves, we recover
Cepheid phases and amplitudes for multi-epoch HST data and we perform template
fitting to derive intensity-averaged mean magnitudes. HST observations in the
SH0ES near-infrared Wesenheit system significantly reduce the effect of
crowding relative to ground-based data, as seen in the final PL scatter of
$\sigma$ = 0.11 mag. We adopt the absolute calibration of the PL based on HST
observations in the Large Magellanic Cloud (LMC) and a distance derived using
late-type detached eclipsing binaries to obtain a distance modulus for M33 of
$\mu$ = 24.622 $\pm$ 0.030 mag (d = 840 $\pm$ 11 kpc), a best-to-date precision
of 1.3%. We find very good agreement with past Cepheid-based measurements.
Several TRGB estimates bracket our result while disagreeing with each other.
Finally, we show that the flux contribution from star clusters hosting Cepheids
in M33 does not impact the distance measurement and we find only 3.7% of the
sample is located in (or nearby) young clusters. M33 offers one of the best
sites for the cross-calibration of many primary distance indicators. Thus, a
precise independent geometric determination of its distance would provide a
valuable new anchor to measure the Hubble constant.
