  We demonstrate that magnetic vortices in which spins are coupled to polar
lattice distortions via superexchange exhibit an unusually large linear
magnetoelectric response. We show that the periodic arrays of vortices formed
by frustrated spins on Kagom\'e lattices provide a realization of this concept;
our {\it ab initio} calculations for such a model structure yield a
magnetoelectric coefficient that is 30 times larger than that of prototypical
single phase magnetoelectrics. Finally, we identify the design rules required
to obtain such a response in a practical material.
