Thanks for writing. It sounds like milkweed vine, a quick growing plant with an extensive root system. If frequent hand-pulling doesn't work (you can eventually exhaust the vine by pulling it out, then watching for regrowth and pulling that as soon as you see it), you may want to consider an herbicide such as Roundup which goes on the plant's foliage and is absorbed into the root system. One way to apply a spray without getting it all over your other plants is to paint it on using a paintbrush dipped in the product.
If you'd prefer not to use synthetic chemicals, you might look around your area and see if you can find organic herbicides; there are getting to be more and more on the market.
---Justin, Senior Garden Editor, BHG.com