You're going to need to do a combination of mechanical and chemical operations IF POSSIBLE. Depending on the severity of the infestation, a chemical may have to be performed after a timber harvest while making sure to have equipment is sprayed thoroughly to prevent/reduce the spread across your property from the equipment.
But if it's a small area...
Cut the vine low to the ground and again at least 4.5 feet higher (creating a gap within the vine). You want to minimize the ability of the vine to climb up and reach itself to cling to easily. Shortly after, spray the bottom section of the vine with (thoroughly wetting the leaves) with one of the following: a glyphosate herbicide (Garlon 3A or Garlon 4) as a 2% solution OR Escort XP at 1-2 ounces per acre in water OR Arsenal AC as a 1% solution. . Ideally, for the most effect, you'll do this in September.
If the vine is largely spread, especially throughout the stand you can wait until after a timber sale and broadcast spray.
CLICK HERE for facts with control recommendations.
"Okay, that seems a bit much."
"I don't believe in herbicides and want the ecosystem to be as natural as possible."
But while that "pretty little vine" is making you "oooohh" and "aahhhh", looking to the left... it will, maybe slowly and gradually, but it will choke out all those beneficial and diverse natural, native plants and wildlife habitat that you are cherishing as a natural ecosystem. While it is flashing it's pretty green dress and copper coins, assuring you there is no urgency to worry about... it will be taking over everything else and robbing you blind of your land's potential.