Add water and sugar to a small pan and bring to a boil over medium heat. Stir until sugar is dissolved, then remove from heat, add 4 of the mint sprigs, and cover for 10 minutes. After 10 minutes, remove the mint sprigs and discard.
Add honeydew, simple syrup, lime juice, and fresh mint leaves from 2 remaining mint sprigs to a high speed blender. Blend until smooth, then pour into a baking sheet and freeze for 1 hour.
After 1 hour, remove from freezer and scrape using a fork or spoon. Spread evenly into pan and freeze for another hour.
Scrape the mixture again using a fork or spoon, then scoop into serving glasses. Serve immediately.