  In this paper, the structure of cocommutative vertex bialgebras is
investigated. For a general vertex bialgebra $V$, it is proved that the set
$G(V)$ of group-like elements is naturally an abelian semigroup, whereas the
set $P(V)$ of primitive elements is a vertex Lie algebra. For $g\in G(V)$,
denote by $V_g$ the connected component containing $g$. Among the main results,
it is proved that if $V$ is a cocommutative vertex bialgebra, then
$V=\oplus_{g\in G(V)}V_g$, where $V_{\bf 1}$ is a vertex subbialgebra which is
isomorphic to the vertex bialgebra ${\mathcal{V}}_{P(V)}$ associated to the
vertex Lie algebra $P(V)$, and $V_g$ is a $V_{\bf 1}$-module for $g\in G(V)$.
  In particular, this shows that every cocommutative connected vertex bialgebra
$V$ is isomorphic to ${\mathcal{V}}_{P(V)}$ and hence establishes the
equivalence between the category of cocommutative connected vertex bialgebras
and the category of vertex Lie algebras. Furthermore, under the condition that
$G(V)$ is a group and lies in the center of $V$, it is proved that
$V={\mathcal{V}}_{P(V)}\otimes \C[G(V)]$ as a coalgebra where the vertex
algebra structure is explicitly determined.
