We propose a new model-agnostic search strategy for hints of new fundamental forces motivated by applications in particle physics. It is based on a novel application of neural density estimation to anomaly detection. Our approach, which we call Classifying Anomalies THrough Outer Density Estimation (CATHODE), assumes potential signal events cluster in phase space in a signal region. However, backgrounds due to known processes are also present in the signal region and too large to directly detect such a signal. By training a conditional density estimator on a collection of additional features outside the signal region, interpolating it into the signal region, and sampling from it, we produce a collection of events that follow the background model. We can then train a classifier to distinguish the data from the events sampled from the background model, thereby approaching the optimal anomaly detector. Using the public LHC Olympics R&D data set, we demonstrate that CATHODE nearly saturates the best possible performance, and significantly outperforms other approaches in this bump hunt paradigm.