In a small saucepan add the Granulated Sugar (1/2 cup) and Water (1/2 cup). Heat to boiling and stir until the sugar dissolves. Set aside to cool.
Trim the Pineapple (1) top and bottom. Cut away the prickly skin, slice the pineapple into quarters, vertically. Cut away the core and discard. Cut the pineapple flesh into chunks.
Transfer them to a heavy duty blender (like a VitaMix). Add simple syrup and the juice from the Lime (1). Puree until smooth. Refrigerate the pineapple mixture until well chilled, about 2 hours.
Freeze according to your ice cream maker's instructions.
Sorbet will be soft, but ready to eat. For firmer sorbet, transfer to a freezer safe container and freeze until firm, at least two hours.