In this work, we present the first results of our imaging campaign at Keck Observatory to identify the host galaxies of ``dark’’ gamma-ray bursts (GRBs), events with no detected optical afterglow or with detected optical flux significantly fainter than expected from the observed X-ray afterglow. We find that out of a uniform sample of 29 Swift bursts rapidly observed by the Palomar 60 inch telescope through 2008 March (14 of which we classify as dark), all events have either a detected optical afterglow, a probable optical host-galaxy detection, or both. Our results constrain the fraction of Swift GRBs coming from very high redshift (z>7), such as the recent GRB 090423, to between 0.2% and 7% at 80% confidence. In contrast, a significant fraction of the sample requires large extinction columns (host-frame A$_V$ gsim 1 mag, with several events showing A$_V$ > 2-6 mag), identifying dust extinction as the dominant cause of the dark GRB phenomenon. We infer that a significant fraction of GRBs (and, by association, of high-mass star formation) occurs in highly obscured regions. However, the host galaxies of dark GRBs seem to have normal optical colors, suggesting that the source of obscuring dust is local to the vicinity of the GRB progenitor or highly unevenly distributed within the host galaxy.