We report on the results of a search for the radio counterpart to the bright gamma-ray burst of March 1, 1994. Using the Dominion Radio Astrophysical Observatory Synthesis Telescope sensitive, wide-field radio images at 1.4 GHz and 0.4 GHz were made of a region around GRB940301. A total of 15 separate radio images were obtained at each frequency, sampling a near-continuous range of post-burst timescales between 3 and 15 days, as well as 26, 47 and 99 days. We place an upper limit of 3.5 mJy on a fading/flaring radio counterpart at 1.4 GHz and 55 mJy at 0.4 GHz. Previous searches have concentrated on searching for a counterpart at only one epoch following the outburst. In contrast, the present search maintains high sensitivity over two decades of post-burst time durations. Time-variable radio emission after the initial gamma-ray burst is a prediction of all fireball models, currently the most popular model for gamma-ray bursts. Our observations allow us to put significant constraints on the fireball parameters for cosmological models of gamma-ray bursts.Comment: 9 pages, postscript. ApJ(Letters) submitted. Hardcopies of paper with figures available on request from dfrail@nrao.edu. Report 00