Butter a 9-inch-round cake pan and line the bottom with parchment paper. Butter the paper.
In a small bowl, whisk the flour, baking powder, baking soda and salt. In a large bowl, using a mixer, beat the butter with the 2/3 cup of sugar at medium-high speed until fluffy, 3 minutes. Beat in the egg and vanilla. At low speed, beat in the buttermilk and dry ingredients in 3 alternating batches, ending with the dry ingredients; do not overbeat. Gently fold the batter just until blended, then scrape into the pan and smooth the top.
Scatter the 1 1/4 cups of blackberries over the batter; lightly press them in.
Sprinkle the remaining 1 1/2 tablespoons of sugar over the cake.
Bake for about 30 minutes, until a cake tester inserted in the center comes out clean.
Transfer the cake to a rack to cool for 10 minutes. Turn out the cake and remove the paper. Turn the cake right side up and let cool completely.