Due to escalating healthcare costs, accurately predicting which patients will incur high costs is an important task for payers and providers of healthcare. High-cost claimants (HiCCs) are patients who have annual costs above $\$250,000$ and who represent just 0.16% of the insured population but currently account for 9% of all healthcare costs. In this study, we aimed to develop a high-performance algorithm to predict HiCCs to inform a novel care management system. Using health insurance claims from 48 million people and augmented with census data, we applied machine learning to train binary classification models to calculate the personal risk of HiCC. To train the models, we developed a platform starting with 6,006 variables across all clinical and demographic dimensions and constructed over one hundred candidate models. The best model achieved an area under the receiver operating characteristic curve of 91.2%. The model exceeds the highest published performance (84%) and remains high for patients with no prior history of high-cost status (89%), who have less than a full year of enrollment (87%), or lack pharmacy claims data (88%). It attains an area under the precision-recall curve of 23.1%, and precision of 74% at a threshold of 0.99. A care management program enrolling 500 people with the highest HiCC risk is expected to treat 199 true HiCCs and generate a net savings of $\$7.3$ million per year. Our results demonstrate that high-performing predictive models can be constructed using claims data and publicly available data alone, even for rare high-cost claimants exceeding $\$250,000$. Our model demonstrates the transformational power of machine learning and artificial intelligence in care management, which would allow healthcare payers and providers to introduce the next generation of care management programs.