We report the spectroscopic detection of mid-infrared emission from the transiting exoplanet HD 209458b. Using archive data taken with the Spitzer/IRS instrument, we have determined the spectrum of HD 209458b between 7.46 and 15.25 microns. We have used two independent methods to determine the planet spectrum, one differential in wavelength and one absolute, and find the results are in good agreement. Over much of this spectral range, the planet spectrum is consistent with featureless thermal emission. Between 7.5 and 8.5 microns, we find evidence for an unidentified spectral feature. If this spectral modulation is due to absorption, it implies that the dayside vertical temperature profile of the planetary atmosphere is not entirely isothermal. Using the IRS data, we have determined the broad-band eclipse depth to be 0.00315 +/- 0.000315, implying significant redistribution of heat from the dayside to the nightside. This work required development of improved methods for Spitzer/IRS data calibration that increase the achievable absolute calibration precision and dynamic range for observations of bright point sources.