We propose the use of a dynamic choice experiment method, which we call Bayesian Adaptive Choice Experiment (BACE), to elicit preferences efficiently. BACE generates an adaptive sequence of menus from which subjects will make choices. Each menu is optimally chosen, according to the mutual information criterion, using the information provided by the subjects' previous choices. We provide sufficient conditions under which BACE achieves convergence and show that its convergence rate significantly improves upon existing discrete choice methods with randomly generated menus. We show that it achieves the highest possible rate of convergence whenever preferences are deterministic. Given that BACE requires the calculation of a Bayesian posterior as well as the solution to a non-trivial optimization problem, several computational challenges arise. We address such challenges by using Bayesian Monte Carlo techniques and provide a package for researchers to employ. The separation between a front-end survey interface and a back-end computational server allows the BACE package to be portable for research designs in a wide range of settings.