Rovibrational quantum states in the X-1 Sigma(+)(g) electronic ground state of H-2 are prepared in the v = 13 vibrational level up to its highest bound rotational level J = 7, and in the highest bound vibrational level v = 14 (for J = 1) by two-photon photolysis of H2S. These states are laser-excited in a subsequent two-photon scheme into F-1 Sigma(+)(g) outer well states, where the assignment of the highest (v,J) states is derived from a comparison of experimentally known levels in F-1 Sigma(+)(g), combined with ab initio calculations of X-1 Sigma(+)(g) levels. The assignments are further verified by excitation of F-1 Sigma(+)(g) population into autoionizing continuum resonances, which are compared with multichannel quantum defect calculations. Precision spectroscopic measurements of the F-X intervals form a test for the ab initio calculations of ground state levels at high vibrational quantum numbers and large internuclear separations, for which agreement is found.