Type II-plateau supernovae (SNe II-P) are the classic variety of core-collapse events that result from isolated, massive stars with thick hydrogen envelopes intact at the time of explosion. Their distances are now routinely estimated through two techniques: the expanding photosphere method (EPM), a primary distance-determining method, and the recently developed standard-candle method (SCM), a promising secondary technique. Using Cycle 10 Hubble Space Telescope (HST) observations, we identify 41 Cepheid variable stars in NGC 1637, the host galaxy of the most thoroughly studied SN II-P to date, SN 1999em. Remarkably, the Cepheid distance that we derive to NGC 1637, D = 11.7 ± 1.0 Mpc, is nearly 50% larger than earlier EPM distance estimates to SN 1999em. This is the first direct comparison between these two primary distance-determining methods for a galaxy hosting a well-observed, spectroscopically and photometrically normal, SN II-P. Extensive consistency checks show strong evidence to support the Cepheid distance scale, so we are led to believe either that SN 1999em is in some heretofore unsuspected way an unusual SN II-P, or that the SN II-P distance scale must be revised. Assuming the latter, this one calibration yields H0(EPM) = 57 ± 15 km s-1 Mpc-1 and H0(SCM) = 59 ± 11 km s-1 Mpc-1; additional calibrating galaxies are clearly desirable in order to test the robustness of both determinations of H0. The HST observations of NGC 1637 also captured the fading SN 1999em 2 yr after explosion, providing the latest photometry ever obtained for an SN II-P. The nebular-phase photometric behavior of SN 1999em closely follows that observed for SN 1987A at similar epochs. The V and I light curves are both declining at rates significantly greater than the decay slope of 56Co predicts. This is likely due to an increasing transparency of the envelope to gamma rays and perhaps also to the formation of dust in the cooling atmosphere of the SN. The absolute V-band brightness of SN 1999em is ~0.25 mag brighter than SN 1987A at the same epochs, which suggests that a slightly greater amount of radioactive 56Ni, ~0.09 M☉, was ejected by SN 1999em than was derived for SN 1987A (0.075 M☉).
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