Theoretical research on the two-dimensional crystal structure of hexagonal boron nitride (h-BN) has suggested that the physical properties of h-BN can be tailored for a wealth of applications by controlling the atomic structure of the membrane edges. Unexplored for h-BN, however, is the possibility that small additional edge-atom distortions could have electronic structure implications critically important to nanoengineering efforts. Here we demonstrate, using a combination of analytical scanning transmission electron microscopy and density functional theory, that covalent interlayer bonds form spontaneously at the edges of a h-BN bilayer, resulting in subangstrom distortions of the edge atomic structure. Orbital maps calculated in 3D around the closed edge reveal that the out-of-plane bonds retain a strong pi* character. We show that this closed edge reconstruction, strikingly different from the equivalent case for graphene, helps the material recover its bulklike insulating behavior and thus largely negates the predicted metallic character of open edges.