We present measurements of 68 classical Cepheids, most detected from the Canada-France-Hawaii Telescope POMME Survey, with periods from 10 to 78 days observed in the near-infrared by the Panchromatic Hubble Andromeda Treasury Program using the Wide Field Camera 3 (WFC3) on the Hubble Space Telescope (HST). The combination of HST's resolution and the use of near-infrared measurements provide a dramatic reduction in the dispersion of the period-luminosity relation over the present optical, ground-based data. Even using random phase magnitudes we measure a dispersion of just 0.17 mag, implying a dispersion of just 0.12 mag for mean magnitudes. The error in the mean for this relation is 1% in distance. Combined with similar observations of Cepheids in other hosts and independent distance determinations, we measure a distance to M31 of μ0 = 24.38 ± 0.06(statistical) ± 0.03(systematic), 752 ± 27 kpc, in good agreement with past measurements though with a better, 3% precision here. The result is also in good agreement with independent distance determinations from two detached eclipsing binaries allowing for an independent calibration of the Cepheid luminosities and a determination of the Hubble constant. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.