  We present results of a study of the central regions of NGC 6397 using Hubble
Space Telescope's Advanced Camera for Surveys, 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 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 Msun. Their
H-alpha--R625 colors indicate that they have strong H-alpha 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 R625 ~ 24.5, well above the magnitude limit of the observations. We
explore the significance of this finding in the context of thick vs. 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 Teff ~ 10,000 K are of the DA class. Finally, we use
the CO WD sequence to measure a distance to the cluster of 2.34 +- 0.13 kpc.
