We present a study of the central regions of NGC 6397 using HST/ACS, focusing on a group of ~24 faint blue stars that form a sequence parallel to, but brighter than, the sequence of carbon-oxygen white dwarfs. We show that these stars, 18 of which are newly discovered, have magnitudes and colors consistent with those of helium-core 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 cluster CVs. The radial distribution of the He WDs is significantly more concentrated than that of either the CO WDs or the turnoff stars. Binary companions are required to explain the implied dynamical masses of >0.8 Msun. We show that the companions are most likely heavy CO WDs and set constraints on the number and characteristics of the cluster binaries needed to explain the He WDs. In contrast to the CO WD sequence, the He WD sequence ends 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.