An important focus of exoplanet research is the determination of the atmospheric temperature structure of strongly irradiated gas giant planets, or hot Jupiters. HD 209458b is the prototypical exoplanet for atmospheric thermal inversions, but this assertion does not take into account recently obtained data or newer data reduction techniques. We reexamine this claim by investigating all publicly available Spitzer Space Telescope secondary-eclipse photometric data of HD 209458b and performing a self-consistent analysis. We employ data reduction techniques that minimize stellar centroid variations, apply sophisticated models to known Spitzer systematics, and account for time-correlated noise in the data. We derive new secondary-eclipse depths of 0.119% ± 0.007%, 0.123% ± 0.006%, 0.134% ± 0.035%, and 0.215% ± 0.008% in the 3.6, 4.5, 5.8, and 8.0 μm bandpasses, respectively. We feed these results into a Bayesian atmospheric retrieval analysis and determine that it is unnecessary to invoke a thermal inversion to explain our secondary-eclipse depths. The data are well fitted by a temperature model that decreases monotonically between pressure levels of 1 and 0.01 bars. We conclude that there is no evidence for a thermal inversion in the atmosphere of HD 209458b.