We present a new probabilistic method for detecting, deblending, and cataloging astronomical objects called the Bayesian Light Source Separator (BLISS). BLISS is based on deep generative models, which embed neural networks within a Bayesian model. For posterior inference, BLISS uses a new form of variational inference known as Forward Amortized Variational Inference (FAVI). FAVI has scaling advantages over Markov chain Monte Carlo and achieves improved fidelity of the posterior approximation compared with traditional variational inference in our application. The BLISS inference routine is fast, requiring a single forward pass of the encoder networks on a GPU once the encoder networks are trained. BLISS can perform fully Bayesian inference on megapixel images in seconds, and produces highly accurate catalogs than traditional methods do. BLISS is highly extensible, and has the potential to directly answer downstream scientific questions in addition to producing probabilistic catalogs.