Cultural traditions have been observed in a wide variety of animal species. It remains unclear, however, what is required for social learning to give rise to stable traditions: what level of precision and what learning strategies are required. We address these questions by fitting models of cultural evolution to learned bird song. We recorded 615 swamp sparrow (Melospiza georgiana) song repertoires, and compared syllable frequency distributions to the output of individual-based simulations. We find that syllables are learned with an estimated error rate of 1.85% and with a conformist bias in learning. This bias is consistent with a simple mechanism of overproduction and selective attrition. Finally, we estimate that syllable types could frequently persist for more than 500 years. Our results demonstrate conformist bias in natural animal behaviour and show that this, along with moderately precise learning, may support traditions whose stability rivals those of humans.