A high pressure, variable temperature NMR probe for low signal to noise relaxation time and self-diffusion coefficient measurements was constructed and tested. The probe has a high pressure limit of 10,000 psia and a current temperature range of 10 to 250 C. Since the magnetic field gradient coils are inside the pressure vessel, the diffusion measurements may be performed with either pulsed or constant gradients. Spin-lattice relaxation times of octane, decane, dodecane, and hexadecane were measured at temperatures between 10 and 85 C and pressures between atmospheric and 6000 psia. These data are the first for these fluids at elevated pressures and represent a significant expansion of the data base. The relaxation rate at constant temperature is proportional to the ratio of the viscosity divided by the temperature, thus being in qualitative agreement with the Bloembergen, Pound, and Purcell (1948) theory. The corresponding states treatment of spin relaxation has been extended to fluids at arbitrary temperature and pressure. Using decane as a reference fluid, the calculated relaxation times of octane, dodecane, and, hexadecane were in good agreement with the experimental values. The low signal to noise capability of the probe was tested by measuring T\sb1 of \sp13CO\sb2 at 504 psia and 25 C. Agreement with the literature value (Smith, 1986) was excellent. The gradient coils were calibrated using benzene. Measurements of benzene at various temperatures, and water at 25 C, gave reliable diffusion coefficients when either the pulsed gradient or the constant gradient method was used.