Preheat the oven to 400 degrees.
Grease a 9-inch round, 1 1/2-inch deep cake pan.
With a fork, stir together the cake flour, cornmeal, baking powder, salt and sugar.
Beat the egg, melted butter, oil and milk in a large mixing bowl until well blended. Add the dry ingredients to the egg mixture and stir until blended. The batter will be somewhat thin and light.
Pour the batter into the prepared pan and bake for about 15 minutes, until a straw inserted in the center comes out with a few moist crumbs. Press the center of the bread; if it feels slightly firm the bread is done. Be careful not to over bake.
Serve warm wedges of the cornbread with butter and honey.