A novel nonpolar structure of 2 monolayer (ML) thick ZnO(0001) films grown on Ag(111) has been revealed, using surface x-ray diffraction and scanning tunneling microscopy. Zn and O atoms are arranged in planar sheets like in the hexagonal boron-nitride prototype structure. The observed depolarization is accompanied by a significant lateral 1.6% expansion of the lattice parameter and a 3% reduced Zn-O bond length within the sheets. The nonpolar structure stabilizes an atomically flat surface morphology unseen for ZnO surfaces thus far. The transition to the bulk wurtzite structure occurs in the 3 - 4 ML coverage range, connected to considerable roughening.