Preheat oven to 350°F. Grease and flour a 9-inch round cake pan.
Combine the flour, baking powder, baking soda, and salt in a medium-sized bowl using a whisk.
Cream the butter and sugar in a big mixing bowl until it becomes light and airy. Add the eggs one at a time, then mix in the vanilla.
Gently incorporate the dry ingredients into the wet ingredients until they are mixed. Gently fold in the blackberries and basil.
Transfer the batter to the ready pan and bake for 35-40 minutes, until a toothpick inserted in the middle comes out clean.
Let the cake sit in the pan for 10 minutes to cool, then move it to a wire rack to cool thoroughly before serving.