We report measurements of the diffusion coefficients of CO2 and N-2 in pure water at temperatures between (298.15 and 423.15) K and pressures between (15 and 45) MPa. The measurements were made by the Taylor dispersion method and have a standard relative uncertainty of 2.3 96. The results were found to be essentially independent of pressure over the range investigated and a simple relation, based on the Stokes Einstein equation, is proposed to correlate the experimental data. Some experimental difficulties arising in the measurement of the diffusivities of slightly soluble acid-gas solutes such as CO2 in water are also discussed.