  In [GMPS] we proved that the moment map image of a $b$-symplectic toric
manifold is a convex $b$-polytope. In this paper we obtain convexity results
for the more general case of non-toric hamiltonian torus actions on
$b$-symplectic manifolds. The modular weights of the action on the connected
components of the exceptional hypersurface play a fundamental role: either they
are all zero and the moment map behaves as in classic symplectic one, or they
are all nonzero and the moment map behaves as in the toric $b$-symplectic case
studied in [GMPS].
