Automated seizure detection from electroencephalography (EEG) would improve the quality of patient care while reducing medical costs, but achieving reliably high performance across patients has proven difficult. Convolutional Neural Networks (CNNs) show promise in addressing this problem, but they are limited by a lack of large labeled training datasets. We propose using imperfect but plentiful archived annotations to train CNNs for automated, real-time EEG seizure detection across patients. While these weak annotations indicate possible seizures with precision scores as low as 0.37, they are commonly produced in large volumes within existing clinical workflows by a mixed group of technicians, fellows, students, and board-certified epileptologists. We find that CNNs trained using such weak annotations achieve Area Under the Receiver Operating Characteristic curve (AUROC) values of 0.93 and 0.94 for pediatric and adult seizure onset detection, respectively. Compared to currently deployed clinical software, our model provides a 31% increase (18 points) in F1-score for pediatric patients and a 17% increase (11 points) for adult patients. These results demonstrate that weak annotations, which are sustainably collected via existing clinical workflows, can be leveraged to produce clinically useful seizure detection models.