  A recent spectrum of the optical afterglow of GRB 970508 suggests that
gamma-ray bursts (GRBs) are cosmological in origin and it is of crucial
importance to derive an accurate distance to each burst. If GRBs occur near
their host galaxies then Lyman limit absorption should be observable in roughly
half the GRB afterglow spectra. Here we outline the methodology to obtain a
redshift from the GRB afterglow spectrum using the recently installed Space
Telescope Imaging Spectrograph (STIS) instrument onboard the Hubble Space
Telescope. A low-resolution spectrum with the Multi-Anode Microchannel Array
(MAMA) detector gives complete spectral coverage over the wavelength range
157.0-318.0 nm (Near UV) and 115.0-174.0 nm (Far UV). Assuming a Target of
Opportunity observation is conducted soon (< 3 weeks) after a bright burst, a
relatively small integration time (about 3 orbits) would be sufficient to
detect the Lyman limit over a wide redshift range (0.3 < z < 2.2). Detection
(or non-detection) of the Lyman limit, in concert with ground-based
observations of nearby galaxies and Mg II and C IV absorption lines, should
provide meaningful constraints on the relationship of GRBs to galaxies.
