This study presents a novel end-to-end architecture that learns hierarchical representations from raw EEG data using fully convolutional deep neural networks for the task of neonatal seizure detection. The deep neural network acts as both feature extractor and classifier, allowing for end-to-end optimization of the seizure detector. The designed system is evaluated on a large dataset of continuous unedited multichannel neonatal EEG totaling 835 hours and comprising of 1389 seizures. The proposed deep architecture, with sample-level filters, achieves an accuracy that is comparable to the state-of-the-art SVM-based neonatal seizure detector, which operates on a set of carefully designed hand-crafted features. The fully convolutional architecture allows for the localization of EEG waveforms and patterns that result in high seizure probabilities for further clinical examination.