We present a deep color-magnitude diagram of the nearby globular cluster NGC 6397, from V and I images obtained with the Wide Field and Planetary Camera 2 aboard the Hubble Space Telescope (HST). A narrow main sequence extending similar to 10 mag below the turnoff is delineated, as well as a white dwarf sequence spanning 3.5 mag. The main sequence has a 1 sigma full width of 0.04 mag down to V-555 similar or equal to 24, and can be distinguished from held stars to a limit of V-555 similar to 26 (M(V) similar to 14). This corresponds to similar to 0.10 M(circle dot), which is close to the hydrogen-burning limit for this low-metallicity cluster. The main-sequence luminosity function rises to a maximum at I-814 similar or equal to 20.5 (M(I) similar or equal to 8.5) and then drops by a factor of similar to 3 over the next 3 mag. The corresponding mass function either rises to the low-mass limit of the observations or flattens out at the very low mass end, depending on the mass-luminosity relation adopted. There is no sign of an end to the main sequence in the present data. The white dwarf sequence contains similar to 40 white dwarfs with M(V) similar or equal to 10.5-14. The brightest similar to 15 white dwarfs form a well-defined sequence that is well matched in position and shape to theoretical cooling sequences for white dwarfs with masses of 0.55 +/- 0.05 M(circle dot), assuming standard values of the distance and reddening of the cluster. The 0.03 mag spread in colors along the upper similar to 2.5 mag of the sequence is consistent with being due to measurement error alone. The intrinsic dispersion in white dwarf masses is no more than 0.05 M(circle dot) and is likely to be significantly smaller. The observed white dwarfs have cooling ages of similar to 0.1-2 Gyr and total numbers that are in keeping with their expected rate of formation given the current rate of evolution past the tip of the red giant branch.