  We report on Hubble Space Telescope (HST) observations of the optical
transient (OT) discovered in the error box of the gamma-ray burst GRB970508.
The object was imaged on 1997 June 2 with the Space Telescope Imaging
Spectrograph (STIS) and Near-Infrared Camera and Multi-Object Spectrometer
(NICMOS). The observations reveal a point-like source with R = 23.1 +- 0.2 and
H = 20.6 +- 0.3, in agreement with the power-law temporal decay seen in
ground-based monitoring. Unlike the case of GRB970228, no nebulosity is
detected surrounding the OT of GRB970508. We set very conservative upper limits
of R ~ 24.5 and H ~ 22.2 on the brightness of any underlying extended source.
If this subtends a substantial fraction of an arcsecond, then the R band limit
is ~25.5. In combination with Keck spectra that show Mg I absorption and [O II]
emission at a redshift of z = 0.835, our observations suggest that the OT is
located in a star-forming galaxy with total luminosity one order of magnitude
lower than the knee of the galaxy luminosity function, L*. Such galaxies are
now thought to harbor the majority of star formation at z ~ 1; therefore, these
observations may provide support for a link between GRBs and star formation.
