Cosmic rays are the most energetic particles arriving at Earth. Although most of them are thought to be accelerated by supernova remnants, the details of the acceleration process and its efficiency are not well determined. Here we show that the pressure induced by cosmic rays exceeds the thermal pressure behind the northeast shock of the supernova remnant RCW 86, where the x-ray emission is dominated by synchrotron radiation from ultrarelativistic electrons. We determined the cosmic-ray content from the thermal Doppler broadening measured with optical spectroscopy, combined with a proper-motion study in x-rays. The measured postshock proton temperature, in combination with the shock velocity, does not agree with standard shock heating, implying that >50% of the postshock pressure is produced by cosmic rays.