We present a record linkage solution that scales to big data volumes and velocities. Our method trains a Siamese deep learning network to encode records so that matching can be done fast and distributed. Compared to the current state-of-the-art methods using similarity functions and blocking, our solution links 100x the data in roughly 60% the time, with comparable precision and recall. We detail the design, training, and implementation of our method, and illustrate model and runtime performance results using a large US physician database and streaming data, implemented using keras/Tensorflow and Spark.