The neutron-inelastic-scattering technique was used to measure the phonon dispersion relations in a high-density crystal of fcc He at 38 K. The crystal was grown at a pressure of 4.93 kbar and a temperature of 38.5 K in a high-pressure sample holder. Its lattice parameter was determined to be 3.915 0.002, equivalent to a molar volume of 9.03 cm3/mol. The measured dispersion curves were found to be in good agreement with a recent calculation by Goldman using the first-order self-consistent phonon theory without short-range correlation functions. The strong anharmonic effects observed in earlier measurements on the crystals of 21 cm3/mol were found to be much less prominent in this He crystal. The magnitude of the multiphonon interference effects on the one-phonon intensities is shown to be less than half of that observed in the low-density crystals. Thermodynamic analysis of the data yielded DM=154 K which indicates that the ratio of mean amplitude of vibration to the nearest-neighbor distance is 8.6%, as opposed to nearly 30% for the lowest-density He crystals. The dependence of the phonon energies on volume is discussed with reference to the earlier work of Traylor et al. on an fcc crystal at 11.7 cm3/mol. Limited measurements were also made at 22 K to determine the temperature dependence of the phonon energies. Unusually large isochoric temperature shifts of as much as 15% for some phonons close to the zone center were found over the range of 22-38 K.