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Ambiguity is a linguistic tool for encoding information efficiently, yet it also causes misunderstandings and disagreements. It is particularly relevant to the domain of misinformation, as fact-checking ambiguous claims is difficult even for experts. In this paper we argue that instead of predicting a veracity label for which there is genuine disagreement, it would be more beneficial to explain the ambiguity. Thus, this work introduces claim disambiguation, a constrained generation task, for explaining ambiguous claims in fact-checking. This involves editing them to spell out an interpretation that can then be unequivocally supported by the given evidence. We collect a dataset of 1501 such claim revisions and conduct experiments with sequence-to-sequence models. The performance is compared to a simple copy baseline and a Large Language Model baseline. The best results are achieved by employing Minimum Bayes Decoding, with a BertScore F1 of 92.22. According to human evaluation, the model successfully disambiguates the claims 72% of the time.