  We generalize the notion of a Rota-Baxter operator on groups and the notion
of a Rota-Baxter operator of weight 1 on Lie algebras and define and study the
notion of a Rota-Baxter operator on a cocommutative Hopf algebra $H$. If
$H=F[G]$ is the group algebra of a group $G$ or $H=U(\mathfrak{g})$ the
universal enveloping algebra of a Lie algebra $\mathfrak{g}$, then we prove
that Rota-Baxter operators on $H$ are in one to one correspondence with
corresponding Rota-Baxter operators on groups or Lie algebras.
