We describe the energy budget of a coronal mass ejection (CME) observed on 1999 May 17 with the Ultraviolet Coronagraph Spectrometer (UVCS). We constrain the physical properties of the CME's core material as a function of height along the corona by using the spectra taken by the single-slit coronagraph spectrometer at heliocentric distances of 2.6 and 3.1 solar radii. We use plasma diagnostics from intensity ratios, such as the O VI doublet lines, to determine the velocity, density, temperature, and non-equilibrium ionization states. We find that the CME core's velocity is approximately 250 km/s, and its cumulative heating energy is comparable to its kinetic energy for all of the plasma heating parameterizations that we investigated. Therefore, the CME's unknown heating mechanisms have the energy to significantly affect the CME's eruption and evolution. To understand which parameters might influence the unknown heating mechanism, we constrain our model heating rates with the observed data and compare them to the rate of heating generated within a similar CME that was constructed by the MAS code's 3D MHD simulation. The rate of heating from the simulated CME agrees with our observationally constrained heating rates when we assume a quadratic power law to describe a self-similar CME expansion. Furthermore, the heating rates agree when we apply a heating parameterization that accounts for the CME flux rope's magnetic energy being converted directly into thermal energy. This UVCS analysis serves as a case study for the importance of multi-slit coronagraph spectrometers for CME studies.