  We show how to carry out the gauging of the Poisson sigma model in an AKSZ
inspired formulation by coupling it to the a generalization of the Weil model
worked out in ref. arXiv:0706.1289 [hep-th]. We call the resulting gauged field
theory, Poisson--Weil sigma model. We study the BV cohomology of the model and
show its relation to Hamiltonian basic and equivariant Poisson cohomology. As
an application, we carry out the gauge fixing of the pure Weil model and of the
Poisson--Weil model. In the first case, we obtain the 2--dimensional version of
Donaldson--Witten topological gauge theory, describing the moduli space of flat
connections on a closed surface. In the second case, we recover the gauged A
topological sigma model worked out by Baptista describing the moduli space of
solutions of the so--called vortex equations.
