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The upcoming next-generation large area radio continuum surveys can expect tens of millions of radio sources, rendering the traditional method for radio morphology classification through visual inspection unfeasible. We present ClaRAN - Classifying Radio sources Automatically with Neural networks - a proof-of-concept radio source morphology classifier based upon the Faster Region-based Convolutional Neutral Networks (Faster R-CNN) method. Specifically, we train and test ClaRAN on the FIRST and WISE images from the Radio Galaxy Zoo Data Release 1 catalogue. ClaRAN provides end users with automated identification of radio source morphology classifications from a simple input of a radio image and a counterpart infrared image of the same region. ClaRAN is the first open-source, end-to-end radio source morphology classifier that is capable of locating and associating discrete and extended components of radio sources in a fast (< 200 milliseconds per image) and accurate (>= 90 %) fashion. Future work will improve ClaRAN's relatively lower success rates in dealing with multi-source fields and will enable ClaRAN to identify sources on much larger fields without loss in classification accuracy., 22 pages, 16 figures, Accepted in Monthly Notices of the Royal Astronomical Society