  The low temperature specific heat of the superconductor MgCNi$_3$ and a
non-superconductor MgC$_{0.85}$Ni$_3$ is investigated in detail. An additional
contribution is observed from the data of MgCNi$_3$ but absent in
MgC$_{0.85}$Ni$_3$, which is demonstrated to be insensitive to the applied
magnetic field even up to 12 Tesla. A detailed discussion on its origin is then
presented. By subtracting this additional contribution, the zero field specific
heat of MgCNi$_3$ can be well described by the BCS theory with the gap ratio
($\Delta/k_BT_c$) determined by the previous tunneling measurements. The
conventional s-wave pairing state is further proved by the magnetic field
dependence of the specific heat at low temperatures and the behavior of the
upper critical field.
