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, 1 1/3 cups water, cucumber, and mint. Blend until smooth. For the smoothest texture, pass the mixture through a fine strainer. 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. Scoop ice into small dishes and garnish with fresh mint sprigs. The ice is best eaten within 24 hours.