Electroencephalography (EEG) is a valuable clinical tool for grading injury caused by lack of blood and oxygen to the brain during birth. This study presents a novel end-to-end architecture, using a deep convolutional neural network, that learns hierarchical representations within raw EEG data. The system classifies 4 grades of hypoxic-ischemic encephalopathy and is evaluated on a multi-channel EEG dataset of 63 hours from 54 newborns. The proposed method achieves a testing accuracy of 79.6% with one-step voting and 81.5% with two-step voting. These results show how a feature-free approach can be used to classify different grades of injury in newborn EEG with comparable accuracy to existing feature-based systems. Automated grading of newborn background EEG could help with the early identification of those infants in need of interventional therapies such as hypothermia.