  We report on Hubble Space Telescope observations of the optical transient
discovered in the error box of GRB970508. The object was imaged on 1997 June 2
with the Space Telescope Imaging Spectrograph and Near-Infrared Camera and
Multi-Object Spectrometer. 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 previous ground-based monitoring. Unlike the case of GRB970228, no
nebulosity is detected surrounding the OT of GRB970508, although Mg I
absorption and [O II] emission seen in Keck spectra at a redshift of z = 0.835
suggest the presence of a dense, star-forming medium. The HST observations 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. Subsequent photometry suggests a
flattening of the light curve at later epochs. Assuming the OT decline follows
a pure power-law and ascribing the flattening to the presence of an underlying
component of constant flux, we find that this must have R = 25.4, consistent
with the upper limits determined by HST. At z = 0.8, this would correspond to
an absolute magnitude in the U band of ~-18, similar to that of the Large
Magellanic Cloud. We propose a scenario in which the host galaxy of the GRB is
of Magellanic type, possibly being a ``satellite'' of one of the bright
galaxies located at few arcseconds from the optical transient.
