We report on the discovery of the radio, infrared, and optical transient coincident with an X-ray transient proposed to be the afterglow of GRB 980703. At later times when the transient has faded below detection, we see an underlying galaxy with R=22.6; this galaxy is the brightest host galaxy (by nearly 2 mag) of any cosmological gamma-ray burst (GRB) thus far. In keeping with an established trend, the GRB is not significantly offset from the host galaxy. Interpreting the multiwavelength data in the framework of the popular fireball model requires that the synchrotron cooling break was between the optical and X-ray bands on 1998 July 8.5 UT and that the intrinsic extinction of the transient is A$_V$=0.9. This is somewhat higher than the extinction for the galaxy as a whole, as estimated from spectroscopy.