Time dilation is one of the most fascinating aspects of special relativity as it abolishes the notion of absolute time. It was first observed experimentally by Ives and Stilwell in 1938 using the Doppler effect. Here we report on a method, based on fast optical atomic clocks with large, but different Lorentz boosts, that tests relativistic time dilation with unprecedented precision. The approach combines ion storage and cooling with optical frequency counting using a frequency comb. 7Li+ ions are prepared at 6.4% and 3.0% of the speed of light in a storage ring, and their time is read with an accuracy of 2×10-10 using laser saturation spectroscopy. The comparison of the Doppler shifts yields a time dilation measurement represented by a Mansouri-Sexl parameter , consistent with special relativity. This constrains the existence of a preferred cosmological reference frame and CPT- and Lorentz-violating `new' physics beyond the standard model.