  Precision measurements are performed on highly excited vibrational quantum
states of molecular hydrogen. The $v=12, J=0-3$ rovibrational levels of H$_2$
($X^1\Sigma_g^+$), lying only $2000$ cm$^{-1}$ below the first dissociation
limit, were populated by photodissociation of H$_2$S and their level energies
were accurately determined by two-photon Doppler-free spectroscopy. A
comparison between the experimental results on $v=12$ level energies with the
best \textit{ab initio} calculations shows good agreement, where the present
experimental accuracy of $3.5 \times10^{-3}$ cm$^{-1}$ is more precise than
theory, hence providing a gateway to further test theoretical advances in this
benchmark quantum system.
