Skim the thick coconut cream from the top of the can of coconut milk and set aside the thinner milk for later.
In a medium saucepan, combine the skimmed cream, a full can of coconut cream, and the curry paste.
Cook over medium heat, stirring constantly, until fully mixed.
Add the reserved thinner coconut milk, corn syrup, sugar, and basil to the saucepan.
Bring the mixture to a simmer over medium heat, then lower the heat, add the fish sauce, and cook, stirring occasionally, for about 10 minutes until slightly thickened.
Use a blender or immersion blender to thoroughly blend the mixture until smooth.
Stir in the lime zest and refrigerate for at least 3 hours or overnight.
Once the mixture is thoroughly chilled, process it in an ice cream maker according to the manufacturer's instructions.
Transfer the ice cream to an airtight container and freeze for about 2 hours.
Remove the ice cream from the freezer 15 minutes before serving to soften slightly.