We present ground-based optical observations of GRB 020124 starting 1.6 hours after the burst, as well as subsequent Very Large Array (VLA) and Hubble Space Telescope (HST) observations. The optical afterglow of GRB 020124 is one of the faintest afterglows detected to ate, and it exhibits a relatively rapid decay, Fν ∝ t−1.60±0.04, followed by further steepening. In addition, a weak radio s ource was found coincident with the optical afterglow. The HST observations reveal that a posit ionally coincident host galaxy must be the faintest host to date,R ∼ > 29.5 mag. The afterglow observations can be explained by severa l models requiring little or no extinction within the host galaxy, Ahost V ≈ 0− 0.9 mag. These observations have significant implications for the statistics of the so-called dark bursts (bursts for whic h no optical afterglow is detected), which are usually attributed to dust extinction within the host galaxy. The fa intness and relatively rapid decay of the afterglow of GRB 020124, combined with the low inferred extinction indic ate that some dark bursts are intrinsically dim and not dust obscured. Thus, the diversity in the underlying pro pe ties of optical afterglows must be observationally determined before substantive inferences can be drawn from the statistics of dark bursts. Subject headings: gamma-rays:bursts — dust:extinction — cosmology:observa tions