We have obtained high-precision interferometric measurements of Vega with the CHARA Array and FLUOR beam combiner in the K' band at projected baselines between 103 and 273 m. The measured visibility amplitudes beyond the first lobe are significantly weaker than expected for a slowly rotating star characterized by a single effective temperature and surface gravity. Our measurements, when compared to synthetic visibilities and synthetic spectrophotometry from a Roche-von Zeipel gravity-darkened model atmosphere, provide strong evidence for the model of Vega as a rapidly rotating star viewed very nearly pole-on. Our best-fitting model indicates that Vega is rotating at 691% of its angular break-up rate with an equatorial velocity of 275 km s super(-1). Together with the measured v sin i, this velocity yields an inclination for the rotation axis of 5. For this model the pole-to-equator effective temperature difference is 62250 K, a value much larger than previously derived from spectral line analyses. A polar effective temperature of 10,150 K is derived from a fit to ultraviolet and optical spectrophotometry. The synthetic and observed spectral energy distributions are in reasonable agreement longward of 140 ran, where they agree to 5% or better. Shortward of 140 ran, the model is up to 10 times brighter than observed. The model has a luminosity of 637 L , a value 35% lower than Vega's apparent luminosity based on its bolometric flux and parallax, assuming a slowly rotating star. Our model predicts the spectral energy distribution of Vega as viewed from its equatorial plane, and it may be employed in radiative models for the surrounding debris disk.