We report the discovery of a radio counterpart to GRB 990123. In contrast to previous well-studied radio afterglows that rise to peak flux on a timescale of a week and then decay over several weeks to months, the radio emission from this gamma-ray burst (GRB) was clearly detected 1 day after the burst, after which it rapidly faded away. The simplest interpretation of this ``radio flare’’ is that it arises from the reverse shock. In the framework of the afterglow models discussed to date, a forward- shock origin for the flare is ruled out by our data. However, at late times, some radio afterglow emission (commensurate with the observed late-time optical emission and the optical afterglow) is expected from the forward shock. The relative faintness of the observed late-time radio emission provides an independent indication of a jetlike geometry in this GRB. We use the same radio observations to constrain two key parameters of the forward shock (the peak flux and peak frequency) to within a factor of 2. These values are inconsistent with the notion advocated by several authors that the prompt optical emission detected by the Robotic Optical Transient Search Experiment smoothly joins the optical afterglow emission. Finally, in hindsight, we now recognize another such radio flare, and this suggests that one out of eight GRBs has a detectable radio flare. This abundance, coupled with the reverse-shock interpretation, suggests that the radio flare phenomenon has the potential to shed new light on the physics of reverse shocks in GRBs.