You don't have as much to do as you think you do... you will negotiate a trade-in value for your vehicle. If the trade in value is more than you owe, the difference will be applied to whatever price you negotiate for the new vehicle. If you owe more than the trade is worth, they can add that into the loan - you'll be borrowing for the amount that you purchase the new vehicle for PLUS the difference between what they'll give you for the trade and what you owe.
You didn't say what year your Mazda3 is, or its condition. Just that you owe about 6720. Prior to going to the dealer, you should call whoever you currently make loan payments to and inquire what the payoff amount is. (The amount you owe.) Keep this in mind when you're discussing a trade-in value for your current vehicle.
However, you won't be the one paying the remaining balance to the bank on your current car. The finance department/bank they work with will pay that balance off (assuming there is a lien on your title.)
edit: Since this appears to be the first time you're trading in a car, you should do plenty of research before heading to a dealer. Does the BMW dealership own a used car lot across the street? If not, realize that they're going to be sending your car to auction. Thus, the amount of money they receive for that car is most likely going to be considerably lower than what a consumer is going to pay for that vehicle.
If they have negotiation room on the price of the vehicle youre interested in, they may offer you a higher value for your trade, intending to make up for it on the price of the vehicle you purchase from them. (Unless that dealer is one of those rare exceptions that the price on the windshield is the exact same price that they'll sell it for to every customer.) If there's very little negotiation room on the price of the vehicle you're interested in, you're going to get a lower trade in value than you may expect.