The single-crystal elastic moduli of alpha-quartz were measured to above 20 GPa in a diamond-anvil cell by Brillouin spectroscopy. The behavior of the elastic moduli indicates that the high-pressure phase transition in quartz is ferroelastic in nature and is driven by softening of C-44 through one of the Born stability criteria. The trends in elastic moduli confirm theoretical predictions, but there are important differences, particularly with respect to the magnitudes of the B-i. The quartz I-II transition occurs prior to complete softening of the mode and amorphization.