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We report an upper limit on the flux of relativistic monopoles based on the nonobservation of in-ice showers by the Radio Ice Cherenkov Experiment (RICE) at the South Pole. We obtain a 95% C.L. limit of order 10-18(cm2 s sr)-1 for intermediate-mass monopoles of 107 ≤ γ ≤ 1012 at the anticipated energy Etot = 1016 GeV. This bound is over an order of magnitude stronger than all previously published experimental limits for this range of boost parameters γ and exceeds 2 orders of magnitude improvement over most of the range. We review the physics of radio detection, describe a Monte Carlo simulation including continuous and stochastic energy losses, and compare to previous experimental limits.