We report on observations of the fading optical counterpart of the gamma-ray burst GRB 970228, made with the Hubble Space Telescope STIS CCD approximately six months after outburst and with the HST/NICMOS and Keck/NIRC approximately one year after outburst. The unresolved counterpart is detected by STIS at V=28.0 +/- 0.25, consistent with a continued power-law decline with exponent -1.14 +/- 0.05. The counterpart is located within, but near the edge of, a faint extended source with diameter ~0."8 and integrated magnitude V=25.8 +/- 0.25. A reanalysis of HST and NTT observations performed shortly after the burst shows no evidence of proper motion of the point source or fading of the extended emission. Only the extended source is visible in the NICMOS images with a magnitude of H=23.3 +/- 0.1. The Keck observations find K = 22.8 +/- 0.3. Several distinct and independent means of deriving the foreground extinction in the direction of GRB 970228 all agree with A_V = 0.75 +/- 0.2. After adjusting for Galactic extinction, we find that the size of the observed extended emission is consistent with that of galaxies of comparable magnitude found in the Hubble Deep Field (HDF) and other deep HST images. Only 2% of the sky is covered by galaxies of similar or greater surface brightness; therefore the extended source is almost certainly the host galaxy. Additionally, we find that the extinction-corrected V-H and V-K colors of the host are as blue as any galaxy of comparable or brighter magnitude in the HDF. Taken in concert with recent observations of GRB 970508, GRB 971214, and GRB 980703 our work suggests that all four GRBs with spectroscopic identification or deep multicolor broad-band imaging of the host lie in rapidly star-forming galaxies.