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HAT-P-20b is a giant metal-rich exoplanet orbiting a metal-rich star. We analyze two secondary eclipses of the planet in each of the 3.6 and 4.5 μm bands of Warm Spitzer. We have developed a simple, powerful, and radically different method to correct the intra-pixel effect for Warm Spitzer data, which we call pixel-level decorrelation (PLD). PLD corrects the intra-pixel effect very effectively, but without explicitly using - or even measuring - the fluctuations in the apparent position of the stellar image. We illustrate and validate PLD using synthetic and real data and comparing the results to previous analyses. PLD can significantly reduce or eliminate red noise in Spitzer secondary eclipse photometry, even for eclipses that have proven to be intractable using other methods. Our successful PLD analysis of four HAT-P-20b eclipses shows a best-fit blackbody temperature of 1134 ± 29 K, indicating inefficient longitudinal transfer of heat, but lacking evidence for strong molecular absorption. We find sufficient evidence for variability in the 4.5 μm band that the eclipses should be monitored at that wavelength by Spitzer, and this planet should be a high priority for James Webb Space Telescope spectroscopy. All four eclipses occur about 35 minutes after orbital phase 0.5, indicating a slightly eccentric orbit. A joint fit of the eclipse and transit times with extant RV data yields and establishes the small eccentricity of the orbit to high statistical confidence. HAT-P-20b is another excellent candidate for orbital evolution via Kozai migration or other three-body mechanisms.