Quantum Monte Carlo calculations of the properties of bulk hydrogen at zero temperature have been performed. The only approximations involved in these calculations are the restriction to finite systems (64 to 432 atoms), the use of the fixed-node approximation to treat Fermi statistics, and the finite length of the Monte Carlo runs. The Born-Oppenheimer approximation was avoided by solving the quantum many-body problem simultaneously both for the electron and proton degrees of freedom. Using different trial functions and several different crystal structures the transition between the explored molecular and atomic phases was determined to occur at 3.0 +- 0.4 Mbar. The transition to a rotationally ordered molecular phase occurred at about 1.0 Mbar. A lower bound to the static dielectric constant, given in terms of the static structure factor, was found to lie close to experimental values and became large for pressures greater than 500 kbar.