Second-epoch HST observations of NGC 6397 have led to the measurement of proper motions accurate enough to separate the faintest cluster stars from the field, thus extending the luminosity function of this globular cluster far enough to approach the limit of hydrogen burning on the main sequence. We isolate a sample of 1385 main-sequence stars, from just below the turnoff down to I=24.5 (M_I~12.5), which corresponds to a mass of less than 0.1 solar masses for the metallicity of this cluster. Below I=22 (M_I~10), the main-sequence luminosity function drops rapidly, in a manner similar to that predicted by theoretical models of low-mass stars near the hydrogen-burning limit. Comment: Tex, 4 pages, 4 figures, 2 style files (apjlet.tex, twocolumns.tex); Astrophysical Journal Letters, accepted