The ACS Virgo Cluster Survey consists ofHST ACS imaging for 100 early-type galaxies in the Virgo Cluster, observed in the F475W (≈SDSSg) and F850LP (≈SDSSz) filters. We derive distances for 84 of these galaxies using the method of surface brightness fluctuations (SBFs), present the SBF distance catalog, and use this database to examine the three-dimensional distribution of early-type galaxies in the Virgo Cluster. The SBF distance moduli have a mean (random) measurement error of 0.07 mag (0.5 Mpc), or roughly 3 times better than previous SBF measurements for Virgo Cluster galaxies. Five galaxies lie at a distance ofd ≈ 23 Mpc and are members of the W' cloud. The remaining 79 galaxies have a narrow distribution around our adopted distance of ⟨d⟩ = 16.5 ± 0.1 (random mean error) ±1.1 Mpc (systematic). The rms distance scatter of this sample is σ(d) = 0.6 ± 0.1 Mpc, with little or no dependence on morphological type or luminosity class (i.e., 0.7 ± 0.1 and 0.5 ± 0.1 Mpc for the giants and dwarfs, respectively). The back-to-front depth of the cluster measured from our sample of early-type galaxies is 2.4 ± 0.4 Mpc (i.e., ±2 σ of the intrinsic distance distribution). The M87 (cluster A) and M49 (cluster B) subclusters are found to lie at distances of 16.7 ± 0.2 and 16.4 ± 0.2 Mpc, respectively. There may be a third subcluster associated with M86. A weak correlation between velocity and line-of-sight distance may be a faint echo of the cluster velocity distribution not having yet completely virialized. In three dimensions, Virgo's early-type galaxies appear to define a slightly triaxial distribution, with axis ratios of (1 : 0.7 : 0.5). The principal axis of the best-fit ellipsoid is inclined ~20°-40° from the line of sight, while the galaxies belonging to the W' cloud lie on an axis inclined by ~10°-15°.