  We study a Poisson structure $\pi$ on the Grothendieck resolution $X$ of a
complex semi-simple group $G$ and prove that the desingularization map
$\mu:(X,\pi) \to (G,\pi_0)$ is Poisson, where $\pi_0$ is a Poisson structure
such that intersections of conjugacy classes and opposite Bruhat cells $BwB_-$
are Poisson subvarieties. We compute the symplectic leaves of $X$ and show that
$(X, \pi)$ resolves singularities of $(G, \pi_0)$.
