Sown hydrogen is predicted(1,2) to become metallic at high pressures. Although metallization was recently reported in high-pressure shock-wave compression experiments using liquid hydrogen(3), efforts to understand the high-pressure behaviour of the solid phase have relied mainly on spectroscopic studies in the diamond-anvil cell(4-6) and on ab initio calculations(7-10). Central to these studies is the high-pressure crystal structure-something that is difficult to determine in the diamond-anvil cell-and its pressure dependence. Here we report X-ray diffraction measurements of the structure of single-crystal molecular hydrogen at pressures of up to 109 GPa for H-2 and 119 GPa for D-2. From these measurements we deduce the high-pressure equation of state (EOS : the volume-pressure relation). We find that solid hydrogen is more compressible than previously thought, that the crystal becomes increasingly anisotropic with pressure, and that the difference in EOS between H-2 and D-2 is unexpectedly small. The ’softening’ of the EOS relative to previous estimates(4) increases by about 25% the expected transition pressure to the atomic phase. Our EOS differs significantly from that predicted by ab initio calculations(7,9), indicating that theoretical understanding of the behaviour of dense hydrogen remains incomplete.