Yes, simple enough.
Phone your current finance provider and ask for a settlement figure.
What essentially happens is that when you buy a new car, the garage you use will "buy" your old car off you for the settlement figure. They pay that direct to the finance company.
Now, if you use something like CarFinance247 (who are very good but very pushy), tell them what that figure is and that you're looking to trade in the car. They tell the garages in advance so everything is in place.
The only downside is if you got your current car recently (say, within the last 2 years). In that case the settlement figure might be higher than the garage is willing to trade in - you'll either need to find the difference or use a lender who will add this difference to your new finance agreement.
You would get more money for your existing car selling privately, but for me the hassle factor is too much.