Preheat the oven to 400°. 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. Serve with whipped cream and more blackberries.