Forecasting how the risk of pathogen spillover changes over space is essential for the effective deployment of interventions such as human or wildlife vaccination. However, due to the sporadic nature of spillover events, developing robust predictions is challenging. Recent efforts to overcome this obstacle have capitalized on machine learning to predict spillover risk. A weakness of these approaches has been their reliance on human infection data, which is known to suffer from strongly biased reporting. We develop a novel approach that combines sub-models for reservoir species distribution, pathogen distribution, and transmission into the human population. We apply our method to Lassa virus, a zoonotic pathogen with a high threat of emergence in West Africa. The resulting model predicts the distribution of Lassa virus spillover risk and allows us to revise existing estimates for the annual number of new human infections. Our model predicts that between 961,300 -- 4,037,400 humans are infected by Lassa virus each year, an estimate that exceeds current conventional wisdom. Our model also predicts that Nigeria accounts for more than half of all new Lassa cases in humans, making it a high-risk area for Lassa virus to become an emergent pathogen.