The far-ultraviolet (FUV) number counts of galaxies constrain the evolution of the star-formation rate density of the universe. We report the FUV number counts computed from FUV imaging of several fields including the Hubble Ultra Deep Field, the Hubble Deep Field North, and small areas within the GOODS-North and -South fields. These data were obtained with the Hubble Space Telescope Solar Blind Channel of the Advance Camera for Surveys. The number counts sample a FUV AB magnitude range from 21-29 and cover a total area of 15.9 arcmin^2, ~4 times larger than the most recent HST FUV study. Our FUV counts intersect bright FUV GALEX counts at 22.5 mag and they show good agreement with recent semi-analytic models based on dark matter merger trees by Somerville et al. (2011). We show that the number counts are ~35% lower than in previous HST studies that use smaller areas. The differences between these studies are likely the result of cosmic variance; our new data cover more lines of sight and more area than previous HST FUV studies. The integrated light from field galaxies is found to contribute between 65.9 +/-8 - 82.6 +/-12 photons/s/cm^2/sr/angstrom to the FUV extragalactic background. These measurements set a lower limit for the total FUV background light.
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