We present the search for Cherenkov signatures from relativistic magnetic monopoles in data taken with the AMANDA-II detector, a neutrino telescope deployed in the Antarctic ice cap at the Geographic South Pole. The non-observation of a monopole signal in data collected during the year 2000 improves present experimental limits on the flux of relativistic magnetic monopoles: Our flux limit varies between 3.8x10(-17) cm(-2) s(-1) sr(-1) (for monopoles moving at the vacuum speed of light) and 8.8x10(-16) cm(-2) s(-1) sr(-1) (for monopoles moving at a speed beta=v/c=0.76, just above the Cherenkov threshold in ice). These limits apply to monopoles that are energetic enough to penetrate the Earth and enter the detector from below the horizon. The limit obtained for monopoles reaching the detector from above the horizon is less stringent by roughly an order of magnitude, due to the much larger background from down-going atmospheric muons. This looser limit is however valid for a larger class of magnetic monopoles, since the monopoles are not required to pass through the Earth.