The dispersion relation $\epsilon \left(k\right)$ of the elementary excitations of superfluid $4\mathrm{He}$ has been measured at very low temperatures, from saturated vapor pressure up to solidification, using a high-flux time-of-flight neutron-scattering spectrometer equipped with a high-spatial-resolution detector (${10}^5$ “pixels”). A complete determination of $\epsilon \left(k\right)$ is achieved, from very low wave vectors up to the end of Pitaeskii's plateau. The results compare favorably in the whole the wave-vector range with the predictions of the dynamic many-body theory. At low wave vectors, bridging the gap between ultrasonic data and former neutron measurements, the evolution with the pressure from anomalous to normal dispersion, as well as the peculiar wave-vector dependence of the phase and group velocities, are accurately characterized. The thermodynamic properties have been calculated analytically, developing Landau's model, using the measured dispersion curve. A good agreement is found below 0.85 K between direct heat capacity measurements and the calculated specific heat, if thermodynamically consistent power series expansions are used. The thermodynamic properties have also been calculated numerically; in this case, the results are applicable with excellent accuracy up to 1.3 K, a temperature above which the dispersion relation itself becomes temperature dependent.