Bring sugar and 1 cup water to a boil in a medium saucepan. Reduce heat, and simmer, stirring occasionally, until sugar has dissolved and mixture is syrupy, about 4 minutes. Let cool completely.
Puree papayas, lime juice, and honey in a food processor. Transfer to a medium bowl. Stir in sugar syrup.
Freeze in an ice cream maker according to manufacturer's instructions. Transfer to an airtight container; freeze at least 2 1/2 hours (up to 1 week). Serve garnished with lime slices.