Argh! I hate this stuff and it's everywhere in my garden. All efforts to eradicate it have failed.
So what does one do to find the solution to a problem? You google it! I found a mention somewhere that someone took the ends of the vine and stuck them in a strong Roundup solution and killed them to the root. Aha!! In talking about it with my DH, we came up with the idea to buy little flower tubes, those things florists use in flower arranging, fill them with Roundup, stick 'em in the ground with the ends of the plant thrust into the cap. Brilliant! :D
These tubes are pointed on the bottom so they easily are pushed into the ground and the cap ensures that the Roundup doesn't spill.
My friends, it is working!! And it works faster if you find the base of the plant, tear off a couple of the largest vines about 9-12 inches from the base, strip back the leaves and stick them in the tube. In an hour or so you'll see that the vine is beginning to shrivel and by the second day the smaller plants are dead totally; I'm giving the larger plants another day. And there's plenty of weed killer left in the tubes to plunk them down in another spot.
Maybe this idea will help you; maybe you've had a brilliant weed killing idea of your own you'd like to share . . . leave me a comment.