  We measure the UV-continuum slope beta for over 4000 high-redshift galaxies
over a wide range of redshifts z~4-8 and luminosities from the HST HUDF/XDF,
HUDF09-1, HUDF09-2, ERS, CANDELS-N, and CANDELS-S data sets. Our new beta
results reach very faint levels at z~4 (-15.5 mag: 0.006 L*(z=3)), z~5 (-16.5
mag: 0.014L*(z=3)), and z~6 and z~7 (-17 mag: 0.025 L*(z=3)). Inconsistencies
between previous studies led us to conduct a comprehensive review of systematic
errors and develop a new technique for measuring beta that is robust against
biases that arise from the impact of noise. We demonstrate, by object-by-object
comparisons, that all previous studies, including our own and those done on the
latest HUDF12 dataset, suffer from small systematic errors in beta. We find
that after correcting for the systematic errors (typically d(beta) ~0.1-0.2)
all beta results at z~7 from different groups are in excellent agreement. The
mean beta we measure for faint (-18 mag: 0.1L*(z=3)) z~4, z~5, z~6, and z~7
galaxies is -2.03+/-0.03+/-0.06 (random and systematic errors),
-2.14+/-0.06+/-0.06, -2.24+/-0.11+/-0.08, and -2.30+/-0.18+/-0.13,
respectively. Our new beta values are redder than we have reported in the past,
but bluer than other recent results. Our previously reported trend of bluer
beta's at lower luminosities is confirmed, as is the evolution to bluer beta's
at high redshifts. beta appears to show only a mild luminosity dependence
faintward of M(UV,AB) ~ -19 mag, suggesting that the mean beta asymptotes to ~
-2.2 to -2.4 for faint z>~4 galaxies. At z~7, the observed beta's suggest
non-zero, but low dust extinction, and they agree well with values predicted in
cosmological hydrodynamical simulations.
