A 1.3% distance to M33 from HST Cepheid photometry

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.