We present Spitzer Space Telescope infrared photometric time series of the transiting extrasolar planet system TrES-1. The data span a predicted time of secondary eclipse, corresponding to the passage of the planet behind the star. In both bands of our observations, we detect a flux decrement with a timing, amplitude, and duration as predicted by published parameters of the system. This signal represents the first direct detection of (i.e., the observation of photons emitted by) a planet orbiting another star. The observed eclipse depths (in units of relative flux) are 0.00066 ± 0.00013 at 4.5 μm and 0.00225 ± 0.00036 at 8.0 μm. These estimates provide the first observational constraints on models of the thermal emission of hot Jupiters. Assuming that the planet emits as a blackbody, we estimate an effective temperature of T_p = 1060 ± 50 K. Under the additional assumptions that the planet is in thermal equilibrium with the radiation from the star and emits isotropically, we find a Bond albedo of A = 0.31 ± 0.14. This would imply that the planet absorbs the majority of stellar radiation incident upon it, a conclusion of significant impact to atmospheric models of these objects. We also compare our data to a previously published model of the planetary thermal emission, which predicts prominent spectral features in our observational bands due to water and carbon monoxide. This model adequately reproduces the observed planet-to-star flux ratio at 8.0 μm; however, it significantly overpredicts the ratio at 4.5 μm. We also present an estimate of the timing of the secondary eclipse, which we use to place a strong constraint on the expression e cos ω, where e is the orbital eccentricity and ω is the longitude of periastron. The resulting upper limit on e is sufficiently small that we conclude that tidal dissipation is unlikely to provide a significant source of energy interior to the planet.