Too many variables in this scenario:
Does the carrier have fully loaded, fueled and manned fighters on the launch deck at the beginning?
Does the battleship have it's main guns and/or secondary guns loaded?
Does the battleship have up-to-date antiaircraft defenses? I'm guessing that, at only a mile out, modern antiaircraft weapons might be able to hit the fighters while they're on the deck of an aircraft carrier. According to Wikipedia, the affective range of the Phalanx defense system is over twice this, which means that any fighters that take off are probably getting taken out before they can successfully target the battleship, although they might not be able to hit the fighters that haven't launched yet.
Are the crews at their battle stations while they wait for the starting bell, or does the alert only sound once the fight begins?
If the carrier is allowed to have fueled/loaded/manned planes in the catapults before the fight starts, it'll probably win, although this might be a tie where both ships sink (the battleship sinks the carrier, and the fighters that got into the air take out the battleship). If both sides start cold, and a mile away from each other, I'm betting on the battleship. It probably takes less time to load, aim and fire the main guns from a battleship than to get fighters onto the flight deck, arm them, fuel, them, and launch them.
Generally speaking though, I'm betting on the battleship if the carrier has been spotted AND is within range of the battleship's guns, and both sides are made aware of each other at the same time. And the battleship's 5-inch guns alone are probably more than enough to devastate the carrier at a mile out.