Cut the kernels from the corn. Using the back of a knife, scrape and press the juices from the cob. Save both the kernels and the liquid.
In a small bowl, mix together 1 tablespoon milk and the cornstarch.
In a large bowl, whisk together the cream cheese and salt.
In a large saucepan over medium high heat, combine the remaining milk, heavy cream, 2/3 cup sugar, corn syrup, corn kernels and corn juices. Bring the mixture to a rolling boil and boil for 4 minutes. Remove from the heat and pour through a fine mesh strainer into a bowl discarding the kernels before returning to the saucepan. Whisk in the milk cornstarch mixture and boil for 1 minute.
Slowly pour the hot milk into the large bowl with the cream cheese and whisk to combine.
Cool the mixture over an ice bath. Refrigerate until cooled completely.
While the mixture is cooling, make the blackberry sauce. In a small saucepan over medium-high heat, bring 1 cup sugar and the blackberries to a boil and cook for 5-8 minutes until the sugar has dissolved and the berries have released their juices. Pour through a fine mesh strainer and discard the seeds. Chill the sauce in the refrigerator.
Churn in an ice cream maker following the manufacturer’s instructions. Transfer to a storage container alternating layers of ice cream and blackberry sauce. Do not swirl. Freeze the ice cream until firm before serving.