Transitions between glassy″ local minima of a model free-energy functional for a dense hard-sphere system are studied numerically using a microcanonical″ Monte Carlo method that enables us to obtain the transition probability as a function of the free energy and the Monte Carlo ``time.″ The growth of the height of the effective free-energy barrier with density is found to be consistent with a Vogel-Fulcher law. The dependence of the transition probability on time indicates that this growth is primarily due to an increase in the difficulty of finding low-free-energy paths to other minima.