The nature of the neutrino masses is an important open question in particle physics; neutrinos were conceived and implemented into the Standard Model as massless, and their masses, now known to be nonzero, represent an area of new physics. Essential to the investigation of this area are mechanisms by which one might measure these masses, including the hypothetical neutrinoless double-beta decay, which has a lifetime related to the neutrino masses. Experiments searching for this rare decay, such as EXO, are impacted heavily by sources of background radiation. Here, a prototype two-phase time projection chamber is described, having superior temporal resolution to the existing EXO architecture. A machine-learning analysis of data from this prototype is used for pulse-shape discrimination, which has the potential to significantly increase EXO's sensitivity; the preliminary efforts described here are able to reduce backgrounds by 94%, while rejecting only 21% of the signal.