We report the results of infrared (8 μm) transit and secondary eclipse photometry of the hot Neptune exoplanet, GJ 436b using Spitzer. The nearly photon-limited precision of these data allows us to measure an improved radius for the planet and to detect the secondary eclipse. The transit (centered at HJD = 2454280.78149 ± 0.00016) shows the flat-bottomed shape typical of infrared transits, and it precisely defines the planet-to-star radius ratio (0.0839 ± 0.0005), independent of the stellar properties. However, we obtain the planetary radius, as well as the stellar mass and radius, by fitting to the transit curve simultaneously with an empirical mass-radius relation for M dwarfs (M = R). We find R_* = M_* = 0.47 ± 0.02 in solar units, and R_p = 27,600 ± 1170 km (4.33 ± 0.18 R_⊕). This radius significantly exceeds the radius of a naked ocean planet and requires a gaseous hydrogen-helium envelope. The secondary eclipse occurs at phase 0.587 ± 0.005, proving a significant orbital eccentricity (e = 0.150 ± 0.012). The amplitude of the eclipse [(5.7 ± 0.8) × 10^-4] indicates a brightness temperature for the planet of T = 712 ± 36 K. If this is indicative of the planet's physical temperature, it suggests the occurrence of tidal heating in the planet. An uncharacterized second planet likely provides ongoing gravitational perturbations that maintain GJ 436b's orbit eccentricity over long timescales.