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Maximum-likelihood decoding is one of the central algorithmic problems in coding theory. It has been known for over 25 years that maximum-likelihood decoding of general linear codes is NP-hard. Nevertheless, it was so far unknown whether maximum-likelihood decoding remains hard for any specific family of codes with nontrivial algebraic structure. In this paper, we prove that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes. We moreover show that maximum-likelihood decoding of Reed-Solomon codes remains hard even with unlimited preprocessing, thereby strengthening a result of Bruck and Naor.