Machine learning models based on neural networks and deep learning are being rapidly adopted for many purposes. What those models learn, and what they may share, is a significant concern when the training data may contain secrets and the models are public -- e.g., when a model helps users compose text messages using models trained on all users' messages. This paper presents exposure: a simple-to-compute metric that can be applied to any deep learning model for measuring the memorization of secrets. Using this metric, we show how to extract those secrets efficiently using black-box API access. Further, we show that unintended memorization occurs early, is not due to over-fitting, and is a persistent issue across different types of models, hyperparameters, and training strategies. We experiment with both real-world models (e.g., a state-of-the-art translation model) and datasets (e.g., the Enron email dataset, which contains users' credit card numbers) to demonstrate both the utility of measuring exposure and the ability to extract secrets. Finally, we consider many defenses, finding some ineffective (like regularization), and others to lack guarantees. However, by instantiating our own differentially-private recurrent model, we validate that by appropriately investing in the use of state-of-the-art techniques, the problem can be resolved, with high utility.