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Load balancing requests across a cluster of back-end servers is critical for avoiding performance bottlenecks and meeting service-level objectives (SLOs) in large-scale cloud computing services. This paper shows how a small, fast popularity-based front-end cache can ensure load balancing for an important class of such services; furthermore, we prove an O(n log n) lower-bound on the necessary cache size and show that this size depends only on the total number of back-end nodes n, not the number of items stored in the system. We validate our analysis through simulation and empirical results running a key-value storage system on an 85-node cluster.