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.
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