Degassing of the Earth is still poorly understood, as is the large scatter in He/Ar ratios observed in mid-ocean ridge basalts. A possible explanation for such observations is that vesiculation occurs at great depths with noble-gas solubilities different from those measured at 1 bar (ref. 1). Here we develop a hard-sphere model for noble-gas solubility and find that, owing to melt compaction, solubility may decrease by several orders of magnitude when pressure increases, an effect subtly overbalanced by the compression of the fluid phase. Our results satisfactorily explain recent experimental data on argon solubility in silicate melts, where argon concentration increases almost linearly with pressure, then levels off at pressures of 50-100 kbar (refs 2-5). We also model vesiculation during magma ascent at ridges and find that noble-gas partitioning between melt and CO2 vesicles at depth differs significantly from that at low pressure. Starting at 10 kbar (approximately 35 km depth), several stages of vesiculation occur followed by vesicle loss, which explains the broad variability of He-Ar concentration data in mid-ocean ridge basalts. 'Popping rocks', exceptional samples with high vesicularity, may represent fully vesiculated ridge magma, whereas common samples would simply have lost such vesicles.