Peel the persimmon, place in a blender or juicer and puree. Reserve. Put the mint and lime juice in a muddler. Muddle just long enough to release the mint oils.
Place the crushed ice in a high ball glass. Add 3 tablespoons persimmon puree, muddled lime juice and mint leaves, simple syrup, and rum. Top with club soda. Stir gently and serve garnished with a lime slice and mint sprig.
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