  We consider a weakly interacting quantum spin chain with random local
interactions. We prove that many-body localization follows from a physically
reasonable assumption that limits the extent of level attraction in the
statistics of eigenvalues. In a KAM-style construction, a sequence of local
unitary transformations is used to diagonalize the Hamiltonian by deforming the
initial tensor product basis into a complete set of exact many-body
eigenfunctions.
