  An anologue of the Calabi invariant for Poisson manifolds is considered. For
any Poisson manifold $P$, the Poisson bracket on $C^{\infty}(P)$ extends to a
Lie bracket on the space $\Omega^{1}(P)$ of all differential one-forms, under
which the space $Z^{1}(P)$ of closed one-forms and the space $B^{1}(P)$ of
exact one-forms are Lie subalgebras. These Lie algebras are related by the
exact sequence: $$0\lon \reals \lon C^{\infty}(P)\stackrel{d}{\lon}
Z^{1}(P)\stackrel{f}{\lon} H^{1}(P, \reals)\lon 0, $$ where $H^{1}(P,\reals)$
is considered as a trivial Lie algebra, and $f$ is the map sending each closed
one-form to its cohomology class. The goal of the present paper is to lift this
exact sequence to the group level for compact Poisson manifolds under certain
integrability condition. In particular, we will give a geometric description of
a Lie group integrating the underlying Poisson algebra $C^{\infty}(P) $.
