A sample of 198 soft X-ray-selected active galactic nuclei (AGN) from the ROSAT International X-ray Optical Survey (RIXOS) is used to investigate the X-ray luminosity function and its evolution. RIXOS, with a flux limit of 3x10^-14 erg s^-1 cm^-2 (0.5 to 2.0 keV), samples a broad range in redshift over 20 deg^2 of sky, and is almost completely identified; it is used in combination with the Einstein Extended Medium Sensitivity Survey (EMSS) to give a total sample of over 600 AGN. We find the evolution of AGN with redshift to be consistent with pure luminosity evolution (PLE) models in which the evolution slows markedly or stops at high redshifts (z<~1.8). We find that this result is not affected by the inclusion, or exclusion, of narrow-emission-line galaxies at low redshift in the RIXOS and EMSS samples, and is insensitive to uncertainties in the conversion between flux values measured with ROSAT and Einstein. We confirm, using a model-independent V_e/V_a test, that our survey is consistent with no evolution at high redshifts.