We present results of a study of the central regions of NGC 6397 using Hubble Space Telescope's Advanced Camera for Surveys (HST ACS), focusing on a group of 24 faint blue stars that form a sequence parallel to, but brighter than, the more populated sequence of carbon-oxygen white dwarfs (CO WDs). Using F625W, F435W, and F658N filters with the Wide Field Channel (WFC), we show that these stars, 18 of which are newly discovered, have magnitudes and colors consistent with those of helium-core white dwarfs (He WDs) with masses ~0.2-0.3 . Their Hα – R ₆₂₅ colors indicate that they have strong Hα absorption lines, which distinguishes them from cataclysmic variables in the cluster. The radial distribution of the He WDs is significantly more concentrated to the cluster center than that of either the CO WDs or the turnoff stars and most closely resembles that of the cluster's blue stragglers. Binary companions are required to explain the implied dynamical masses. We show that the companions cannot be main-sequence stars and are most likely heavy CO WDs. The number and photometric masses of the observed He WDs can be understood if ~1%-5% of the main-sequence stars within the half-mass radius of the cluster have white dwarf companions with orbital periods in the range ~1-20 days at the time they reach the turnoff. In contrast to the CO WD sequence, the He WD sequence comes to an end at R ₆₂₅ 24.5, well above the magnitude limit of the observations. We explore the significance of this finding in the context of thick versus thin hydrogen envelope models and compare our results to existing theoretical predictions. In addition, we find strong evidence that the vast majority of the CO WDs in NGC 6397 down to T _eff 10,000 K are of the DA class. Finally, we use the CO WD sequence together with theoretical cooling models to measure a distance to the cluster of 2.34 ± 0.13 kpc, or (m – M)₆₂₅ = 12.33.