We discuss a mechanism for the realization of supersolids in lattices in the absence of intersite interactions that surprisingly works as well at unit filling. This mechanism, that we study for the case of the sawtooth lattice, is based on the existence of frustrated and unfrustrated plaquettes. For sufficiently large interactions and frustration the particles gather preferentially at unfrustrated plaquettes breaking spontaneously translational invariance, resulting in a supersolid. We show that for the sawtooth lattice the supersolid exists for a large region of parameters for densities above half filling. Our results open a feasible path for realizing supersolids in existing ultracold atomic gases in optical lattices without the need for long-range interactions. © 2015 American Physical Society.