We accurately determine a new Cepheid distance to M101 (NGC 5457) using archival Hubble Space Telescope (HST)/Advanced Camera for Surveys V and I time series photometry of two fields within the galaxy. We make a slight modification to the ISIS image subtraction package to obtain optimal differential light curves from HST data. We discovered 827 Cepheids with periods between 3 and 80 days, the largest extragalactic sample of Cepheids observed with HST by a factor of two. With this large Cepheid sample, we find that the relative distance of M101 from the Large Magellanic Cloud is Delta *D Delta *mLMC = 10.63 ? 0.04 (random) ? 0.06 (systematic) mag. If we use the geometrically determined maser distance to NGC 4258 as our distance anchor, the distance modulus of M101 is Delta *m0 = 29.04 ? 0.05 (random) ? 0.18 (systematic) mag or D = 6.4 ? 0.2 (random) ? 0.5 (systematic) Mpc. The uncertainty is dominated by the maser distance estimate (?0.15 mag), which should improve over the next few years. We determine a steep metallicity dependence, Delta *g, for our Cepheid sample through two methods, yielding Delta *g = --0.80 ? 0.21 (random) ? 0.06 (systematic) mag dex--1 and Delta *g = --0.72+0.22 -- 0.25 (random) ? 0.06 (systematic) mag dex--1. We see marginal evidence for variations in the Wesenheit period-luminosity relation slope as a function of deprojected galactocentric radius. We also use the tip of the red giant branch method to independently determine the distance modulus to M101 of Delta *m0 = 29.05 ? 0.06 (random) ? 0.12 (systematic) mag.