In a small saucepan, heat the water and sugar to a gentle simmer. Cook until the sugar dissolves, about 4 minutes. Sprinkle the gelatin over the simmering syrup and whisk constantly over low heat until the gelatin dissolves, about 1 minute.
Put the syrup, lime juice and mint leaves in a heatproof bowl. Let stand 15 minutes.
Stir in rum. Strain into a large measuring cup. Squeeze the mint leaves to extract as much liquid (and flavor) as possible. Stir well.
Arrange 30 small paper cups on a sheet tray. Divide the mixture equally among the cups, then refrigerate until chilled and firm, about 3 hours.