Preheat the oven to 425 degrees F and position a rack in the middle. Butter an 8-inch-square baking pan.
Melt the butter in the microwave in a microwave-safe dish in three 15-second intervals on high or in a small pan on the stove. Set it aside to cool.
In a bowl, whisk together the flour, cornmeal, baking powder, baking soda and salt.
In a large bowl, whisk together the buttermilk and the eggs. Add the melted butter. Add the flour-cornmeal mixture and stir just until combined. Pour the batter into the pan. Bake until the cornbread just begins to brown and a toothpick inserted in the center comes out clean, 18 to 23 minutes. Cool for about 10 minutes before serving.