HAT-P-30-WASP-51b is a hot-Jupiter exoplanet that orbits an F star every 2.8106 days at a distance of 0.0419 AU. Using the Spitzer Space Telescope in 2012 (Spitzer Program Number 70084) we observed two secondary eclipses at 3.6 and 4.5 μm. We present eclipse-depth measurements of 0.177 ± 0.018 % and 0.247 ± 0.024 % and estimate the infrared brightness temperatures to be 1990 ± 110 K and 2080 ± 130 K for these two channels, respectively, from an analysis using our Photometry for Orbits, Eclipses, and Transits (POET) pipeline. These may be grazing eclipses. We also refine its orbit using our own secondary-eclipse measurements in combination with radial- velocity and transit observations from both professional and amateur observers. Using only the phase of our secondary eclipses, we can constrain e cos(ω) where e is the orbital eccentricity and ω is the argument of periastron to 0.0058 ± 0.00094. This is the component of eccentricity in the plane of view,. This small but non-zero eccentricity is independent of the effects that stellar tides have on radial-velocity data. When including radial velocity data in our model, our Markov chain finds an e cos(ω) of 0.0043 ± 0.0007. We constrain the atmospheric temperature profile using our Bayesian Atmospheric Radiative Transfer code (BART), a large lower bound (700 km) for the scale height, and the potential for high quality transit spectroscopy observations.