Grate the zest (colored part only) off 2 of the limes and scrape into a large bowl. (If the zest is in large pieces, chop it finely.) Juice the limes, measure 3/4 cup and pour it in with the zest. Add the sugar, corn syrup and 1 3/4 cups water. Stir until the sugar dissolves. Pour the mixture into the canister of your ice cream maker and freeze according to the manufacturer’s directions.
The ice will have the best texture if you scoop it from the ice cream maker into a container and let it firm up for several hours in the freezer. The ice is best eaten within 24 hours, scooped into small dishes and sprinkled with the berries.