What do you wish to prevent? I assume both the unwanted reseeding and the inferior offspring. I realize that it sounds labor intensive, but to prevent the presumed perennializing you must deadhead your spent blossoms in order to keep the plants from dropping every seed that they wish. And to halt and prevent the degradation of your strain you must selectively choose seeds from plants with characteristics of the original that you wish to preserve. Just because a plant is an heirloom and "breeds true" doesn't mean that it won't degrade left to its own devices.
Yank out everything except for the strongest, and collect the seed from the best of those. Also, if you if there are other morning glories in the area they can potentially taint the original strain.