We present Space Telescope Imaging Spectrograph observations of the optical transient (OT) counterpart of the c-ray burster GRB 000301C obtained 5 days after the burst, on 2000 March 6. CCD clear-aperture imaging reveals a R similar or equal to 21.50 +/- 0.15 source with no apparent host galaxy. An 8000 s, 1150 Angstrom < <2200 near-ultraviolet MAMA prism spectrum shows a flat or slightly rising continuum (in f(lambda)) between 2800 and 3300 Angstrom, with a mean flux of (8.7(-1.6)(+0.8) +/- 2.6) x 10(-18) ergs s(-1) cm(-2) Angstrom (-1), and a sharp break centered at 2797 +/- 25 Angstrom. We interpret this as the H I Lyman break at z = 2.067 +/- 0.025, indicating the presence of a cloud with an H I column density log N-HI(cm(2)) > 18 on the line of sight to the OT. This measured redshift is conservatively a lower limit to the GRB redshift. However, as all other GRBs that have deep Hubble Space Telescope images appear to lie on the stellar field of a host galaxy, and as the large H I column density measured here and in later ground-based observations is unlikely on a random line of sight, we believe we are probably seeing absorption from H I in the host galaxy. In any case, this represents the largest direct redshift determination of a c-ray burster to date. Our data are compatible with an OT spectrum represented by a power law with an intrinsic index alpha = 1.2 (f(nu) proportional to nu (-alpha)) and no extinction in the host galaxy, or with alpha = 0.5 and extinction by SMC-like dust in the OT rest frame with A(V) = 0.15. The large N-HI and the lack of a detected host are similar to the situation for damped Ly alpha absorbers at z > 2.