1. In a nonreactive bowl, combine the watermelon, lime juice, and 3 tablespoons sugar. Let sit for 1 hour to draw out juices.

2. Drain off the accumulated juice and combine it with the remaining 3 tablespoons sugar in a small saucepan. Heat over a low flame until sugar is dissolved; do not boil. Let cool, then puree the sugar syrup, watermelon meat, and any additional accumulated juice in a food processor. Transfer to a bowl and add the corn syrup, stirring so that it doesn't accumulate at the bottom of the bowl. Chill, then pour into an ice-cream maker and freeze according to manufacturer's instructions.