An equilibrated polycrystalline material is considered, containing a small fraction of void space (melt or vapor), for which the crystals are above their roughening transition against this medium. The grain boundaries are approximated to have equal free energy per unit area, so that they assume the geometry of lamellae in a dry foam. It is argued that the void space of the polycrystal consists of sharp-edged cavities of tetrahedral symmetry at the grain vertices, and universal shapes for these voids are calculated. Treating these voids as incipient fractures, a general analytic expression for the failure stress (by fracture) of the polycrystal is derived, in terms of void fraction, grain size and polydispersity, fracture toughness and surface and grain boundary free energies. This expression is expected to apply when grains are large enough for fracture to be brittle, and the relation of this mechanism to other modes of material failure, such as plastic yield (as described by the Hall–Petch relation) or breakage under stresses from elastic anisotropy is discussed.