It has recently been reported, on the basis of extrapolated experimental data, that the iron carbide, Fe7C3, has shear wave velocities and a Poisson's ratio consistent with the seismological values for the Earth's inner core and thus that Fe7C3 is a strong candidate for the inner core composition. In this study, using ab initio molecular dynamics simulations, we report the thermoelastic properties of Fe7C3 at 350 GPa up to its melting temperature. Due to significant elastic softening prior to melting, the calculated elastic properties, including wave velocities, do indeed agree well with those from seismology. However, the density was found to be much too low (by ~8%) when compared to geophysical data, and therefore, Fe7C3 must be ruled out as a major component of the Earth's inner core.