Combine the sugar and 1 cup water in a small stainless steel pot and heat it gently, stirring, until the sugar is dissolved. Add the basil and a pinch of salt and simmer gently for 10 minutes. Turn off the heat and let the syrup cool completely. Strain the syrup and reserve the basil. You should have 1 1/4 cups syrup.

Put the peaches in a blender, add the cooled syrup and the lemon juice, and puree. Add about 1 teaspoon of the reserved basil and pulse a few times, just until the basil makes green flecks in the peach puree. For a more pronounced basil flavor, add a teaspoon or so of finely chopped raw basil.

Chill the mixture well, then freeze it in an ice cream maker, following the manufacturer’s instructions. Transfer the sorbet to a chilled container, cover tightly and put it in the freezer for about 2 hours to firm up.