Preheat oven to 350 degrees. Butter and flour a 13 3/4-by-4 1/4-inch fluted tart pan with a removable bottom, and set aside. Combine the rhubarb and 1/3 cup sugar in a medium bowl, and allow to sit, stirring occasionally, until the rhubarb has released its juice and the sugar has dissolved, about 45 minutes.
Sift the flour, baking powder, and salt into a small bowl; set aside. Cream the butter with 2/3 cup sugar in the bowl of an electric mixer fitted with the paddle attachment. Add the eggs, one at a time, beating well after each addition. Slice vanilla bean in half lengthwise. Scrape seeds from pod, and add seeds to the mixture. Beat to combine. Add reserved flour mixture, and beat to combine. Using a rubber spatula, spoon batter into the prepared pan, and spread flat.
Strain rhubarb, discarding juice; add blackberries, and toss to combine. Spoon fruit on top of the batter in the pan, sprinkle with remaining tablespoon sugar, place in the oven, and bake until the cake is golden and center is set, about 1 hour. Remove from the oven, and place on a wire rack to cool slightly before removing tart pan.