We imaged the X-ray error box of GRB 980703, beginning 22.5 hr after the gamma-ray event, in both the optical R and near-infrared H bands. A fading optical/IR object was detected within the X-ray error box, coincident with the variable radio source reported by Frail et al. in 1998, who also detected the optical transient independently of us. Further imagery revealed the gamma-ray burst (GRB) host galaxy, with R=22.49+/-0.04 and H=20.5+/-0.25, the brightest so far detected. When excluding its contribution to the total flux, both the R- and H-band light curves are well fit by a power-law decay with index alpha~=1.4. Our data suggest an intrinsic column density in the host galaxy of ~3.5x10^21 cm^-2, which indicates the existence of a dense and gas-rich medium in which the GRB occurred, thus supporting the hypernova model scenarios.