The propagation of sound in a shallow water environment is characterized by boundary reflections from the sea surface and sea floor. These reflections result in multiple (indirect) sound propagation paths, which can degrade the performance of passive sound source localization methods. This paper proposes the use of convolutional neural networks (CNNs) for the localization of sources of broadband acoustic radiated noise (such as motor vessels) in shallow water multipath environments. It is shown that CNNs operating on cepstrogram and generalized cross-correlogram inputs are able to more reliably estimate the instantaneous range and bearing of transiting motor vessels when the source localization performance of conventional passive ranging methods is degraded. The ensuing improvement in source localization performance is demonstrated using real data collected during an at-sea experiment.