We propose to use quantum interferences to improve the accuracy of the measurement of the free-fall acceleration g¯ of antihydrogen in the gravitational behavior of antihydrogen at rest (GBAR) experiment. This method uses most antiatoms prepared in the experiment and it is simple in its principle, as interferences between gravitational quantum states are read out without transitions between them. We use a maximum likelihood method for estimating the value of g¯ and assess the accuracy of this estimation by a Monte Carlo simulation. We find that the accuracy is improved by approximately three orders of magnitude with respect to the classical timing technique planned for the current design of the experiment.