  The discrete Ces\`aro operator $\mathsf{C}$ is investigated in the class of
power series spaces $\Lambda_0(\alpha)$ of finite type. Of main interest is its
spectrum, which is distinctly different when the underlying Fr\'echet space
$\Lambda_0(\alpha)$ is nuclear as for the case when it is not. Actually, the
nuclearity of $\Lambda_0(\alpha)$ is characterized via certain properties of
the spectrum of $\mathsf{C}$. Moreover, $\mathsf{C}$ is always power bounded,
uniformly mean ergodic and, whenever $\Lambda_0(\alpha)$ is nuclear, also has
the property that the range $(I-\mathsf{C})^m(\Lambda_0(\alpha))$ is closed in
$\Lambda_0(\alpha)$, for each $m\in\mathbb{N}$.
