We study differentially private mean estimation in a high-dimensional setting. Existing differential privacy techniques applied to large dimensions lead to computationally intractable problems or estimators with excessive privacy loss. Recent work in high-dimensional robust statistics has identified computationally tractable mean estimation algorithms with asymptotic dimension-independent error guarantees. We incorporate these results to develop a strict bound on the global sensitivity of the robust mean estimator. This yields a computationally tractable algorithm for differentially private mean estimation in high dimensions with dimensionindependent privacy loss. Finally, we show on synthetic data that our algorithm significantly outperforms classic differential privacy methods, overcoming barriers to high-dimensional differential privacy. * Equal contribution. Author order determined by a random coin flip of a virtual Canadian Loonie.Preprint. Under review.