  Unlike for $\ell_p$, $1<p\leq\infty$, the discrete Ces\`aro operator $C$ does
not map $\ell_1$ into itself. We identify precisely those weights $w$ such that
$C$ does map $\ell_1(w)$ continuously into itself. For these weights a complete
description of the eigenvalues and the spectrum of $C$ are presented. It is
also possible to identify all $w$ such that $C$ is a compact operator in
$\ell_1(w)$. The final section investigates the mean ergodic properties of $C$
in $\ell_1(w)$. Many examples are presented in order to supplement the results
and to illustrate the phenomena that occur.
