  We model the low-temperature specific heat of solid $^4$He in the hexagonal
closed packed structure by invoking two-level tunneling states in addition to
the usual phonon contribution of a Debye crystal for temperatures far below the
Debye temperature, $T < \Theta_D/50$. By introducing a cutoff energy in the
two-level tunneling density of states, we can describe the excess specific heat
observed in solid hcp $^4$He, as well as the low-temperature linear term in the
specific heat. Agreement is found with recent measurements of the temperature
behavior of both specific heat and pressure. These results suggest the presence
of a very small fraction, at the parts-per-million (ppm) level, of two-level
tunneling systems in solid $^4$He, irrespective of the existence of
supersolidity.
