We study the thin-shell limit of the nonlinear elasticity model for vesicles introduced in part I. We consider vesicles of width 2eps ↓ 0 with elastic energy of order eps^3. In this regime, we show that the limit model is a bending theory for generalized hypersurfaces -- namely, codimension 1 oriented varifolds without boundary. Up to a positive factor, the limit functional is the Willmore energy. In the language of Gamma -convergence, we establish a compactness result, a lower bound result and the matching upper bound in the smooth case.