The spectrum and point spectrum of the Cesàro averaging operator \(\mathsf {C}\) acting on the Fréchet space \(C^\infty ({\mathbb R}_+)\) of all \(C^\infty \) functions on the interval \([0,\infty )\) are determined. We employ an approach via \(C_0\)-semigroup theory for linear operators. A spectral mapping theorem for the resolvent of a closed operator acting on a locally convex space is established; it constitutes a useful tool needed to establish the main result. The dynamical behaviour of \(\mathsf {C}\) is also investigated.