Using ab initio non-Born-Oppenheimer simulations, we demonstrate the amplification of XUV radiation in a high-harmonic-generation-type process using the example of the hydrogen molecular ion. A small fraction of the molecules is pumped to a dissociative excited state from which IR-assisted XUV amplification is observed. We show that starting at sufficiently high IR driving field intensities, the ground-state molecules become quasitransparent for XUV radiation, while due to stabilization, gain from excited states is maintained. While the basic physics should also be observable in atomic media, the main advantage of the investigated molecular laser is, first, efficient lasing from field-free excited states with a high mean angular momentum and, second, the possibility to tune the amplified XUV frequency windows via control of the internuclear distance.