I assume you are talking about one of the annual vines of the morning glory family which are often called Bindweeds. THese sprout from seed each year and can grow very quickly. THere is no chemical to kill weeds but not ground covers unless the ground cover is a graminoid (grass, sedge, etc). I think that constant weeding to prevent any seed from being created is the only choice except removing the ground cover and the weeds (perhaps with Roundup) and starting over.

I have recently learned that people inject roundup into the stems of bindweed with a syringe to kill it. Have yet to try this myself but it seems like it might be a fun afternoon.

The injection thing is new to me too! I go with the old Q-tip dipped in Roundup technique. I have successfully eradicated bindweed from my lawn (dab, dab, dab). It took 3 seasons & constant vigilence, & yes, there is the occasional 1 that shoots up. It gets dabbed immediately. I'm hoping by next summer that there won't be any left. B4 2005's eradication war, my front lawn was more bindweed than grass. So, yes, it's possible to win. But, it requires constant vigilence & immediate action. Instead of putting the roundup on the whole vine, you could try cutting the bindweed off at soil level and dabbing roundup on the cut end. That worked very well for me. Also, use the roundup for brush. Don't get discouraged, some weeds need repeated applications. Eventually, the monster will die.

Binder weed is little understood by most gardeners. I brought a sample to a nursery owner who misdiagnosed it as kudzu. Because I was growing morning glories,at first I thought it might be in the morning glory family because its flower looks very similar to the morning glory in color and shape. The leaves are also similar. (some people However, Binder Weed has roots that winter over which can descend one to two feet underground below the frost line. It is one of the earlier plants coming up in the spring. If you rototill the area you end up with hundreds of new plants. The only solution if chemicals applied to the leaves. Be sure it kills kudzu- If so it will work on binder weed.