Preheat your oven to 400°F (204°C). Grease a 9x9 inch baking dish.
In a large bowl, whisk together the flour, cornmeal, sugar, baking powder, baking soda, and salt.
In another bowl, combine the buttermilk and beaten eggs. Add the melted butter, stirring until combined.
Pour the wet ingredients into the dry ingredients, stirring just until blended. Do not overmix.
Pour the batter into the prepared baking dish, using a spatula to spread it evenly.
Bake for 20-25 minutes, until the cornbread is golden brown and a toothpick inserted into the center comes out clean.
Allow the cornbread to cool in the dish for 10 minutes before slicing and serving.