Let A be a co-Poisson Hopf algebra with Poisson co-bracket $delta$. Here it is shown that the Hopf dual $A^{circ}$ is a Poisson Hopf algebra with Poisson bracket {f, g}(x) = < $delta(x)$, $f;{otimes};g$ > for any f, g $in$ $A^{circ}$ and x $in$ A if A is an almost normalizing extension over the ground field. Moreover we get, as a corollary, the fact that the Hopf dual of the universal enveloping algebra U(g) for a finite dimensional Lie bialgebra g is a Poisson Hopf algebra.