  For a finitely-generated vertex operator algebra of central charge c, a
locally convex topological completion is constructed. We construct on the
completion a structure of an algebra over the operad of the c/2-th power of the
determinant line bundle over the moduli space of genus-zero Riemann surfaces
with ordered analytically parametrized boundary components. In particular, the
completion is a module for the semi-group of the c/2-th power of the
determinant line bundle over the moduli space of conformal equivalence classes
of annuli with analytically parametrized boundary components. The results in
Part I for Z-graded vertex algebras are also reformulated in terms of the
framed little disk operad. Using May's recognition principle for double loop
spaces, one immediate consequence of such operadic formulations is that the
compactly generated spaces corresponding to (or the k-ifications of) the
locally convex completions constructed in Part I and in the present paper have
the weak homotopy types of double loop spaces. We also generalize the results
above to locally-grading-restricted conformal vertex algebras and to modules.
