We report the first two-dimensional angular correlation of positron-annihilation radiation (ACPAR) studies of the positron-electron momentum distribution for NiO. The data contain two components: a large, isotropic, corelike electron contribution and a remaining valence-electron contribution. We access the covalency structure by modeling the anisotropy with a linear combination of atomic orbitals--molecular orbital method and a localized ion scheme, within the independent-particle model. We obtain the experimental two-dimensional electron-positron k-space momentum density and find no Fermi surface but significant, small residual variations. The importance and implications of these results for ACPAR studies of the new high-${T}_{c}$ perovskites are discussed.