We report Spitzer/IRAC photometry of the transiting giant exoplanet HAT-P-1b during its secondary eclipse. This planet lies near the postulated boundary between the pM and pL-class of hot Jupiters, and is important as a test of models for temperature inversions in hot Jupiter atmospheres. We derive eclipse depths for HAT-P-1b, in units of the stellar flux, that are: 0.080% +/- 0.008%,[3.6um], 0.135% +/- 0.022%,[4.5um],0.203% +/- 0.031%,[5.8um], and $0.238% +/- 0.040%,[8.0um]. These values are best fit using an atmosphere with a modest temperature inversion, intermediate between the archetype inverted atmosphere (HD209458b) and a model without an inversion. The observations also suggest that this planet is radiating a large fraction of the available stellar irradiance on its dayside, with little available for redistribution by circulation. This planet has sometimes been speculated to be inflated by tidal dissipation, based on its large radius in discovery observations, and on a non-zero orbital eccentricity allowed by the radial velocity data. The timing of the secondary eclipse is very sensitive to orbital eccentricity, and we find that the central phase of the eclipse is 0.4999 +/- 0.0005. The difference between the expected and observed phase indicates that the orbit is close to circular, with a 3-sigma limit of |e cosw| < 0.002.