  We present a large scale exact diagonalization study of the one dimensional
spin $1/2$ Heisenberg model in a random magnetic field. In order to access
properties at varying energy densities across the entire spectrum for system
sizes up to $L=22$ spins, we use a spectral transformation which can be applied
in a massively parallel fashion. Our results allow for an energy-resolved
interpretation of the many body localization transition including the existence
of an extensive many-body mobility edge. The ergodic phase is well
characterized by Gaussian orthogonal ensemble statistics, volume-law
entanglement, and a full delocalization in the Hilbert space. Conversely, the
localized regime displays Poisson statistics, area-law entanglement and non
ergodicity in the Hilbert space where a true localization never occurs. We
perform finite size scaling to extract the critical edge and exponent of the
localization length divergence.
