In a small saucepan over medium heat, whisk sugar, cocoa powder, 1/4 cup water, and the corn syrup. Cook, whisking, until mixture begins to boil, then whisk 1 minute more. Remove from heat; stir in vanilla and salt. Transfer to a bowl and chill until very cold (it will be thin, like maple syrup).
Spoon thin layer of ice cream into a 3- or 4-cup storage container. Spoon on a layer of chocolate sauce. Top with another layer of ice cream, then chocolate sauce, and continue to fill container (you may have some sauce left). Freeze until hard, at least 2 hours (sauce will not harden completely).
*If ice cream is airy, you may need as much as 4 cups, because it will collapse when measured.
Make ahead: Chocolate sauce, up to 3 weeks, chilled (thin with a bit of water if needed). Ice cream, up to 3 months, frozen airtight.
Makes about 3 cups (serving size: 1/2 cup)