We report on the discovery of the radio afterglow from the gamma-ray burst of 1999 May 6 (GRB 990506) using the Very Large Array. The radio afterglow was detected at early times (Δt=1.5 days) but began to fade rapidly sometime between 1 and 5 days after the burst. If we attribute the radio emission to the forward shock from an expanding fireball, then this rapid onset of the decay in the radio predicts that the corresponding optical transient began to decay between 1 and 5 minutes after the burst. This could explain why no optical transient for GRB 990506 was detected in spite of numerous searches. The cause of the unusually rapid onset of the decay for the afterglow is probably the result of an isotropically energetic fireball expanding into a low-density circumburst environment. At the location of the radio afterglow, we find a faint (Råisebox-0.5ex 24 mag) host galaxy with a double morphology. The observations reported herein were obtained with the Very Large Array, a facility of the National Radio Astronomy Observatory, which is operated by Associated Universities, Inc., under a cooperative agreement with the National Science Foundation, and with the W. M. Keck Observatory, which is operated by the California Association for Research in Astronomy, a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration.