Preheat the oven to 350°. Butter a 9-by-13-inch baking dish. In a bowl, combine the flour, cornmeal, sugar, salt, baking powder and baking soda. In another bowl, combine the eggs and buttermilk and add to the dry ingredients. Add the melted butter and stir just until moistened.
Pour the batter into the prepared dish and bake for about 40 minutes, or until the corn bread is golden and a toothpick inserted in the center comes out clean. Let cool in the pan for 10 minutes, then turn out onto a wire rack to cool completely.