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Although artificial intelligence (AI) systems have been shown to improve the accuracy of initial melanoma diagnosis, the lack of transparency in how these systems identify melanoma poses severe obstacles to user acceptance. Explainable artificial intelligence (XAI) methods can help to increase transparency, but most XAI methods are unable to produce precisely located domain-specific explanations, making the explanations difficult to interpret. Moreover, the impact of XAI methods on dermatologists has not yet been evaluated. Extending on two existing classifiers, we developed an XAI system that produces text and region based explanations that are easily interpretable by dermatologists alongside its differential diagnoses of melanomas and nevi. To evaluate this system, we conducted a three-part reader study to assess its impact on clinicians' diagnostic accuracy, confidence, and trust in the XAI-support. We showed that our XAI's explanations were highly aligned with clinicians' explanations and that both the clinicians' trust in the support system and their confidence in their diagnoses were significantly increased when using our XAI compared to using a conventional AI system. The clinicians' diagnostic accuracy was numerically, albeit not significantly, increased. This work demonstrates that clinicians are willing to adopt such an XAI system, motivating their future use in the clinic.