Bring the sugar, water, mint, and lemon balm to a simmer in a saucepan over medium heat. Stir until the sugar has dissolved. Strain into a heatproof container. Stir in the muscat wine. Refrigerate until cold, about 4 hours.
Stir in the lemon juice; pour the chilled mixture into an ice cream maker and freeze according to manufacturer's directions until it reaches "soft-serve" consistency. Transfer ice cream to a one- or two-quart plastic container with a lid; cover surface with plastic wrap and seal. For best results, ice cream should ripen in the freezer for at least 2 hours or overnight.