Peel off lime zest using a vegetable peeler to create long strands.
Combine confectioners' sugar, water, and lime zest in a saucepan and bring to a boil. Boil for 5 minutes. Remove from heat and let cool to room temperature, about 30 minutes. Refrigerate simple syrup until completely cold, about 1 hour. Remove and discard lime zest and stir in rum.
Place mango in a blender; blend until pureed. Chill in the refrigerator.
Combine mango puree and chilled simple syrup in an ice cream maker and freeze according to manufacturer's instructions until it reaches soft-serve consistency, about 20 minutes. Transfer sorbet to an airtight container and freeze until firm, about 2 hours.
