Establishing the phase diagram of hydrogen is a major challenge for experimental and theoretical physics. Experiment alone cannot establish the atomic structure of solid hydrogen at high pressure, because hydrogen scatters X-rays only weakly. Instead, our understanding of the atomic structure is largely based on density functional theory (DFT). By comparing Raman spectra for low-energy structures found in DFT searches with experimental spectra, candidate atomic structures have been identified for each experimentally observed phase. Unfortunately, DFT predicts a metallic structure to be energetically favoured at a broad range of pressures up to 400 GPa, where it is known experimentally that hydrogen is non-metallic. Here we show that more advanced theoretical methods (diffusion quantum Monte Carlo calculations) find the metallic structure to be uncompetitive, and predict a phase diagram in reasonable agreement with experiment. This greatly strengthens the claim that the candidate atomic structures accurately model the experimentally observed phases.