We present measurements of the specific ultraviolet luminosity density from a sample of 483 galaxies at 6 . z . 8. These galaxies were selected from new deep near-infrared Hubble Space Telescope imaging from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey, Hubble UltraDeep Field 2009 and WFC3 Early Release Science programs. In contrast to the majority of previous analyses, which assume that the distribution of galaxy ultraviolet (UV) luminosities follows a Schechter distribution, and that the distribution continues to luminosities far below our observable limit, we investigate the contribution to reionization from galaxies which we can observe, free from these assumptions. Due to our larger survey volume, wider wavelength coverage, and updated assumptions about the clumping of gas in the intergalactic medium (IGM), we find that the observable population of galaxies can sustain a fully reionized IGMat z = 6, if the average ionizing photon escape fraction (fesc) is ∼30%. A number of previous studies have measured UV luminosity densities at these redshifts that vary by a factor of 5, with many concluding that galaxies could not complete reionization by z = 6 unless a large population of galaxies fainter than the detection limit were invoked, or extremely high values of fesc were present. The observed UV luminosity density from our observed galaxy samples at z = 7 and 8 is not sufficient to maintain a fully reionized IGM unless fesc > 50%. We examine the contribution from galaxies in different luminosity ranges, and find that the sub-L∗ galaxies we detect are stronger contributors to the ionizing photon budget than the L > L∗ population, unless fesc is luminosity dependent. Combining our observationswith constraints on the emission rate of ionizing photons from Lyα forest observations at z = 6, we find that we can constrain fesc < 34% (2σ) if the observed galaxies are the only contributors to reionization, or < 13% (2σ) if the luminosity function extends to a limiting magnitude of MUV = -13. These escape fractions are sufficient to complete reionization by z = 6. Current constraints on the high-redshift galaxy population imply that the volume ionized fraction of the IGM, while consistent with unity at z ≤ 6, appears to drop at redshifts not much higher than 7, consistent with a number of complementary reionization probes. If faint galaxies dominated the ionizing photon budget at z = 6–7, future extremely deep observations with the James Webb Space Telescope will probe deep enough to see them, providing an indirect constraint on the global ionizing photon escape fraction.