I live in Dallas and I have a very prolific vine that grows through by St. Augustine and also into my flower beds. It does not go away as my grass gets healthier. I have recently realized that it does better in wet areas. It is almost like a succulant. When you hand pull it you never get all of it and it has roots at every joint. At times it has white flowers. I have hand pulled it by the bushels and it comes back just as full.
How do I get rid of it?

Have you sprayed it with full strength vinegar? It will kill the St Augustine but the grass will come back. Sometimes the weed does, to, so you might have to spray weekly.

Can you post a picture of it?

Maple1, what you describe sounds like bindweed. Look it up in the Dirt Doctor Library section to see pictures. It is horribly invasive and has gotten much worse in north central Texas the last few years. But I sure would NOT use anything that would kill the grass just to get rid of bindweed. That would be like burning a building down to get rid of roaches, in my opinion, because once the grass is dead, there sure wouldn't be anything to slow down the bindweed, which loves bare bad soil. Personally, I don't want a yard with nothing green but bindweed. LOL. The healthy part of our yard where we've consistently used cornmeal, molasses, green sand & lava sand mixture since 2001 has no bindweed or fire ants. The other areas where we have not improved the soil yet has bindweed and lots of fire ants.

If you have bind weed, then I agree with funny farm. That's great advice! Are the white flowers shaped like trumpets?