The Space Telescope Imaging Spectrograph (STIS) has measured the absolute flux for Vega from 0.17-1.01 μm on the Hubble Space Telescope white dwarf (WD) flux scale. These data are saturated by up to a factor of 80 overexposure but retain linearity to a precision of 0.2% because the charge bleeds along the columns and is recovered during readout of the CCD. The signal-to-noise ratio per pixel exceeds 1000, and the resolution R is about 500. A V magnitude of 0.026+/-0.008 is established for Vega, and the absolute flux level is 3.46×10-9 ergs cm-2 s-1 at 5556 Å. In the regions of Balmer and Paschen lines, the STIS equivalent widths differ from the pioneering work of Hayes in 1985 but do agree with predictions of a Kurucz model atmosphere, so that the STIS flux distribution is preferred to that of Hayes. Over the full wavelength range, the model atmosphere calculation shows excellent agreement with the STIS flux distribution and is used to extrapolate predicted fluxes into the IR region. However, the IR fluxes are 2% lower than the standard Vega model of Cohen. IUE data provide the extension of the measured STIS flux distribution from 0.17 down to 0.12 μm. The STIS relative flux calibration is based on model atmosphere calculations of pure hydrogen WDs, while the Hayes flux calibration is based on the physics of laboratory lamps and blackbody ovens. The agreement to 1% of these two independent methods for determining the relative stellar flux distributions suggests that both methods may be correct from 0.5-0.8 μm and adds confidence to claims that the fluxes relative to 5500 Å are determined to better than 4% by the pure hydrogen WD models from 0.12 to 3 μm.