1. In a small saucepan, combine the sugar with the 1/4 cup of water and the mint and bring to a boil. Cover, remove from the heat and let stand for 15 minutes. Strain the mint syrup into a large bowl and let cool to room temperature. Whisk in the lime zest, lime juice, gin and cucumber puree.
2. In a microwave-safe bowl, sprinkle the gelatin over the remaining 1 tablespoon of water and let stand until softened, 5 minutes. Microwave the mixture at high power for 8 seconds, until the gelatin dissolves. Whisk the gelatin into the cucumber mixture.
3. Arrange ten 2-ounce paper cups on a baking sheet. Pour the popsicle mixture into the cups and freeze for 1 hour, until partially frozen. Insert a stick in the center of each cup and freeze for 30 minutes longer, until fully frozen. Tear off the paper and serve.