We design a novel, communication-efficient, failure-robust protocol for secure aggregation of high-dimensional data. Our protocol allows a server to collect an aggregate of user-held data from mobile devices in a privacy-preserving manner, and can be used, for example, in a federated learning setting, to aggregate user-provided model updates for a deep neural network. We prove the security of our protocol in the honest-but-curious and malicious server settings, and show that privacy is preserved even if an arbitrarily chosen subset of users drop out at any time. We evaluate the efficiency of our protocol and show, by complexity analysis and a concrete implementation, that its runtime and communication overhead remain low even on large data sets and client pools. For 16-bit input values, our protocol offers 1.73× communication expansion for 2^10 users and 2^20-dimensional vectors, and 1.98× expansion for 2^14 users and 2^24-dimensional vectors.