Learning to predict rare events from sequences of events with categorical features is an important, real-world, problem. Unfortunately, most machine learning methods that learn classification rules are not suited to solving this type of problem because they assume an unordered set of examples and cannot identify patterns between examples (i.e., events). Statistical time-series prediction methods are also not suitable, since they assume numerical features. Genetic algorithms, however, which have often been used to find patterns in data, are well suited to finding predictive temporal and sequential patterns in the event sequence data. In order to solve the event prediction problem, we developed Timeweaver, a genetic-based machine learning system that, given a pre-specified target event, learns to identify patterns in the data that successfully predict the future occurrence of that event. Timeweaver has been applied to the task of predicting telecommunication failures from timestamped alarm messages and has outperformed several simple prediction methods.