We report on the superconducting (SC) properties of Y\(_2\)C\(_3\) with a relatively high transition temperature \(T_{\rm c}=15.7\) K investigated by \(^{13}\)C nuclear-magnetic-resonance (NMR) measurements under a magnetic field. The \(^{13}\)C Knight shift has revealed a significant decrease below \(T_{\rm c}\), suggesting a spin-singlet superconductivity. From an analysis of the temperature dependence of the nuclear spin-lattice relaxation rate \(1/T_1\) in the SC state, Y\(_2\)C\(_3\) is demonstrated to be a multigap superconductor that exhibits a large gap \(2\Delta/k_{\rm B}T_{\rm c}=5\) at the main band and a small gap \(2\Delta/k_{\rm B}T_{\rm c}=2\) at other bands. These results have revealed that Y\(_2\)C\(_3\) is a unique multigap s-wave superconductor similar to MgB\(_2\).