  Consider a Hamiltonian action of a compact connected Lie group $G$ on an
aspherical symplectic manifold $(M,\omega)$. Under suitable assumptions,
counting gauge equivalence classes of (symplectic) vortices on the plane $R^2$
conjecturally gives rise to a quantum deformation $Qk_G$ of the Kirwan map.
  This is the second of a series of articles, whose goal is to define $Qk_G$
rigorously. The main result is that every sequence of vortices with uniformly
bounded energies has a subsequence that converges to a genus 0 stable map of
vortices on $R^2$ and holomorphic spheres in the symplectic quotient.
  Potentially, the map $Qk_G$ can be used to compute the quantum cohomology of
many symplectic quotients. Conjecturally it also gives rise to quantum
generalizations of non-abelian localization and abelianization.
