Arrange melon wedges in a single layer in two 8-by-8 baking dishes.

Combine powdered sugar, lime juice, mint and rum in a blender with 1/2 cup water. Pulse until mint is finely chopped. Pour mixture over melon wedges. You will need to divide the mixture evenly between the two baking dishes. Refrigerate for one hour, flipping wedges after 30 minutes.

Line a baking sheet with parchment paper. Carefully skewer each wedge with a wooden pop stick or skewer and arrange in a single layer on the baking sheet. Freeze for 30 minutes. Serve cold.