Molecular hydrogen in the X 1 Sigma g+ (vX=0,J=1) state was resonantly excited to the E,F 1 Sigma g+ (vE=0,J=1) state by two-photon absorption. The use of a laser with low pulse energy and narrow bandwidth reduced the importance of the photoionisation of this state by the absorption of another photon, and thereby enhanced the role of collisional quenching. The molecular ions and vacuum-ultraviolet fluorescence that resulted from the laser pulse were measured for hydrogen densities from 1015 to 1019 cm-3. An analysis of the rates of excitation and ionisation, and a comparison of the spectrum of the fluorescence with a computed simulation, suggest that bound states in the outer well of the double-well E,F potential were involved in the excitation transfer to the excited states of B 1 Sigma u+ that produce the fluorescence. Neutral and electron collisions, ionic processes and higher-order radiative interactions are discussed as possible explanations of this phenomenon.