Preheat the oven to 375 F.
To make the filling: Whisk together the sugar, cornstarch, cinnamon and salt in a large bowl. Add the blackberries to the bowl and very gently toss to coat with the sugar mixture. Drizzle with the lemon juice. Pour the filling into a 12- x 8-inch baking dish. Bake for 25 minutes, or until the filling is hot and bubbling around the edges.
To make the topping: Whisk together the flour, sugar, cornmeal, baking powder, baking soda and salt. With a pastry blender or a fork, cut in the butter until it is the size of small peas. Stir in the buttermilk until no dry pockets remain.
Remove the cobbler from the oven and drop 6 spoonful's of topping on top of the filling, using all of the topping. Do not let the topping mounds touch. Sprinkle them lightly with sugar. Bake for 15 to 18 minutes, until the filling is hot and bubbly and the topping is brown. Cool on a wire rack for about 20 minutes. Serve warm, topped with a scoop of vanilla ice cream.