  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 H$_2$S. 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, combined with \emph{ab initio} calculations
of \X\ levels. The assignments are further verified by excitation of
$F^1\Sigma_g^+$ population into autoionizing continuum resonances which are
compared with multi-channel quantum defect calculations. Precision
spectroscopic measurements of the $F-X$ intervals form a test for the \emph{ab
initio} calculations of ground state levels at high vibrational quantum numbers
and large internuclear separations, for which agreement is found.
