  We use high quality echelle spectra of 24 QSOs to provide a calibrated
measurement of the total amount of Lyman-alpha forest absorption (DA) over the
redshift range 2.2 < z < 3.2. Our measurement of DA excludes absorption from
metal lines or the Lyman-alpha lines of Lyman limit systems and damped
Lyman-alpha systems. We use artificial spectra with realistic flux calibration
errors to show that we are able to place continuum levels that are accurate to
better than 1%. When we combine our results with our previous results between
1.6 < z < 2.2, we find that the redshift evolution of DA is well described over
1.6 < z < 3.2 as A*(1+z)^gamma, where A = 0.0062 and gamma = 2.75. We detect no
significant deviations from a smooth power law evolution over the redshift
range studied. We find less H I absorption than expected at z=3, implying that
the UV background is about 40% higher than expected. Our data appears to be
consistent with an H I ionization rate of Gamma ~ 1.4E-12 1/s.
