For the syrup, bring the sugar and water to a boil. Cool the syrup and add the rum.
For the pasta bignè, bring the water, butter, and salt to a boil in a small saucepan. Remove from the heat and sift in the flour, all at once. Stir until smooth and return to the heat. Cook, stirring, about 30 seconds. Pour the paste into a bowl and beat in the eggs, one at a time. Pour the paste into a pastry bag fitted with a ¼-inch plain tube (Ateco #2) and pipe the paste in ⅜-inch spheres onto two parchment-lined cookie sheets. Bake the bignè at 375 degrees for about 15 minutes, until well puffed and deep golden. Cool the bignè on the sheets.
For the chocolate filling, bring the cream to a boil, remove from the heat, and add the chocolate. Allow to stand 2 minutes, then whisk until smooth. Pour into a bowl and cool. Beat the butter until soft and light. Beat in the cooled chocolate mixture, beating until light. Beat in the rum, a little at a time, being careful not to overbeat, which will make the filling separate.
To assemble the cake, cut two ¼-inch slices from the layer, using a long, serrated knife. Place one slice on a cardboard cake circle and reserve the other one, covered with plastic wrap. Cut the remainder of the cake into 3 layers, then trim one, to a 9-inch-round layer, the second to an 8-inch-round layer, and the third to a 7-inch-round layer. Reserve the scraps, covered with plastic wrap. Moisten the layer on the cardboard lightly with the syrup, then spread with one-eighth of the filling. Place the 9-inch-round layer on the filling and moisten with the syrup. Spread with another eighth of the filling. Place the 8-inch-round layer on the filling, moisten with the syrup, and spread with another eighth of the filling. Place the 7-inch-round layer on and moisten. Spread with another eighth of the filling. Use the scraps to form a rough 6-inch-round layer on the filling. Moisten and spread with another eighth of the filling. Use more scraps to form a 5-inch-round layer; moisten and spread with another eighth of the filling. Use the last scraps to form a 4-inch-round layer. Moisten it and spread the entire outside of the cake with another eighth of the filling. Make a cut in the reserved layer, from the center outward to the edge, and arrange it smoothly, on the stacked dome of layers, overlapping it at the cut. Moisten with the last syrup and spread with the remaining filling. Press the bignè all over the outside of the cake so they adhere.
Chill briefly to let the filling set. Dust lightly with the confectioners’ sugar, then with the cocoa powder. Keep the cake at a cool room temperature until served. Store leftovers in the refrigerator. Bring to room temperature before serving.