NGC 6397 is the second closest globular star cluster to the Sun. Using 5 days of time on the Hubble Space Telescope, we have constructed an ultradeep color-magnitude diagram for this cluster. We see a clear truncation in each of its two major stellar sequences. Faint red main-sequence stars run out well above our observational limit and near to the theoretical prediction for the lowest mass stars capable of stable hydrogen burning in their cores. We also see a truncation in the number counts of faint blue stars, namely white dwarfs. This reflects the limit to which the bulk of the white dwarfs can cool over the lifetime of the cluster. There is also a turn toward bluer colors in the least luminous of these objects. This was predicted for the very coolest white dwarfs with hydrogen-rich atmospheres as the formation of H2 and the resultant collision-induced absorption cause their atmospheres to become largely opaque to infrared radiation.