1) Heat 1 cup of the watermelon puree with the sugar, whisking until the sugar dissolves.
2) Add the rest of the puree, the lime and the corn syrup and whisk to combine.
3) Take the puree off the heat and add the Campari.
4) If you have an ice cream maker, pour the mixture in and proceed according to its instructions. Otherwise, pour the mixture into a plastic container and freeze for 4 hours (or overnight).
5) Let the frozen mixture stand at room temp for a few minutes to soften slightly, then process (or blend) and freeze again.