In a bowl, mash 1/3 cup sugar and lime peel with a spoon to release oils. Let stand 5 to 10 minutes.
Stir 1/2 cup lime juice into sugar mixture. Pour mixture through a fine strainer into a blender or food processor; discard lime peel.
Add melon, a portion at a time, and whirl until smooth. Return purée to bowl. Taste, and if desired, add more sugar and lime juice. Nest bowl in ice and stir often until mixture is cold, 5 to 10 minutes.
Pour chilled purée into an ice cream maker (1-qt. or larger capacity). Freeze according to manufacturer's directions until sorbet is firm to scoop, dasher is hard to turn, or machine stops.
Serve, or freeze airtight.
Firming and Storing Ice Cream:
To get frozen desserts hard enough to scoop onto a cone, or to store them, transfer when frozen to an airtight container and put in the freezer at least 3 hours or up to 1 week.
If freezing with ice and salt, leave the frozen dessert in ice and salt up to 3 hours.
For best flavor and texture, serve frozen desserts within a week. On longer standing, icy crystals develop.
Nutritional analysis per 1/2 cup.