My guess is bindweed. I read an interesting method for controlling this noxious weed on this forum...you leave the plant in the ground, gather up a bunch of the foliage and stuff it into a jar with some concentrated Roundup. You need to protect the contents from rain and secure it so it doesn't topple. The plant will take in the herbicide and this will kill it, roots and all.

Yes, it's bindweed.

Thanks a lot Donna and Jean! I will give the method Donna mentioned a try - it sounds like a good attacking plan.