  The Poisson--Weil sigma model, worked out by us recently, stems from gauging
a Hamiltonian Lie group symmetry of the target space of the Poisson sigma
model. Upon gauge fixing of the BV master action, it yields interesting
topological field theories such as the 2--dimensional Donaldson-Witten
topological gauge theory and the gauged A topological sigma model. In this
paper, generalizing the above construction, we construct the Lie algebroid
Poisson sigma model. This is yielded by gauging a Hamiltonian Lie groupoid
symmetry of the Poisson sigma model target space. We use the BV quantization
approach in the AKSZ geometrical version to ensure consistent quantization and
target space covariance. The model has an extremely rich geometry and an
intricate BV cohomology, which are studied in detail.
