At first I was startled by this question, not being a morning-glory grower and not having heard of an infestation of said plant before. But as soon as I looked around the web…WOW. As with any infestation, you have your work cut out for you. And your weapons are the same (none perfect): A hoe in the early going, thick sheets of plastic in areas where it gets going and you want to cook/smother it, and most of all any tactic you can muster to prevent more flowers from maturing and setting more seed. In other words, welcome to another uphill battle, not unlike my life with Houttuynia and all the other issues people have had with theirs unwanted roommates (see piece below for some tactics others are using, in comments). http://awaytogarden.com/why-wont-this-plant-die Sounds like this will take years, but I have seen patient, determined friends eradicate whole yards of goutweed and persistence can pay off. Sorry not to have a miracle cure. Did you, by the way, reach out to Seed Savers Exchange, which I recall was the original source of this plant, to see if they have any insight?