Place a 10-inch cast-iron skillet in oven, and preheat oven to 450°F. Heat skillet 7 minutes.
Meanwhile, stir together the cornmeal, flour, sugar, baking powder, baking soda, and salt in a large bowl. Stir together the buttermilk and eggs in a medium bowl.
Add butter to hot skillet, and return to oven until butter is melted, about 1 minute. Stir buttermilk mixture into cornmeal mixture until just combined. Pour melted butter from skillet into cornmeal mixture, and quickly stir to incorporate. Pour mixture into hot skillet, and immediately place in oven.
Bake in preheated oven until golden brown and cornbread pulls away from sides of skillet, 18 to 20 minutes. Remove from skillet; cool slightly before serving.