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Advancing the utility of social media data for research applications requires methods for automatically detecting demographic information about social media study populations, including users' age. The objective of this study was to develop and evaluate a method that automatically identifies the exact age of users based on self-reports in their tweets. Our end-to-end automatic natural language processing (NLP) pipeline, ReportAGE, includes query patterns to retrieve tweets that potentially mention an age, a classifier to distinguish retrieved tweets that self-report the user's exact age ("age" tweets) and those that do not ("no age" tweets), and rule-based extraction to identify the age. To develop and evaluate ReportAGE, we manually annotated 11,000 tweets that matched the query patterns. Based on 1000 tweets that were annotated by all five annotators, inter-annotator agreement (Fleiss' kappa) was 0.80 for distinguishing "age" and "no age" tweets, and 0.95 for identifying the exact age among the "age" tweets on which the annotators agreed. A deep neural network classifier, based on a RoBERTa-Large pretrained transformer model, achieved the highest F1-score of 0.914 (precision = 0.905, recall = 0.942) for the "age" class. When the age extraction was evaluated using the classifier's predictions, it achieved an F1-score of 0.855 (precision = 0.805, recall = 0.914) for the "age" class. When it was evaluated directly on the held-out test set, it achieved an F1-score of 0.931 (precision = 0.873, recall = 0.998) for the "age" class. We deployed ReportAGE on a collection of more than 1.2 billion tweets, posted by 245,927 users, and predicted ages for 132,637 (54%) of them. Scaling the detection of exact age to this large number of users can advance the utility of social media data for research applications that do not align with the predefined age groupings of extant binary or multi-class classification approaches.