We measure the amount of absorption in the Lyα forest at 0 < z < 1.6 in Hubble Space Telescope Faint Object Spectrograph spectra of 74 QSOs. Starting with a 334 QSO sample compiled by Bechtold et al. 2002, we selected 74 QSOs that have the highest signal to noise and complete coverage of rest frame wavelengths 1070 –1170 Å. We measure the absorption from the flux in each pixel in units of the unabsorbed continuum level. We mask out regions of spectra that contain metal lines, or strong Lyα lines that are accompanied by other Lyman series line or metals at the same redshift, leaving Lyα absorption from the low density intergalactic medium. At 0 < z < 1.6 we find that 79% of the absorption is from the low density intergalactic medium, 12% from metals and 9% from the strong H I lines, nearly identical to the percentages (78, 15 and 7) that we measured independently at z = 2 from spectra taken with the Kast spectrograph on the Lick 3-m. At z = 1 the low density intergalactic medium absorbs 0.037± 0.004 of the flux. The error includes some but not all of the uncertainty in the continuum level. The remaining part gives relative errors of approximately 0.21 when we report the mean absorption in eight independent redshift intervals, and 0.047 when we average over all redshifts. We find 1.46 times more absorption from the low density intergalactic medium than comes from Lyα lines that Bechtold et al. 2002 listed in the same spectra. The amount of absorption increases with z and can be fit by a power law in (1 + z) with index 1.01. This corresponds to no change in the number of lines, of fixed rest frame equivalent widths, per unit redshift, consistent with the Janknecht et al. 2006 results on the distribution of lines. When we include similar measurements from higher redshifts, we need more degrees of freedom to fit the amount of absorption at 0 < z < 3.2. A power law with a break in slope, changing from index 1.5 at low z to 3.0 above z ∼ 1.1 is a better but only marginally acceptable fit. We also calculate two other continuous statistics, the flux probability distribution function and the flux autocorrelation function that is non-zero out to v ∼ 500 km s at 0.5 < z < 1.5.