Without seeing a photo of the vine, we can't be certain what the plant is. However, from your description, I suspect that you might have bindweed, which is related to morning glory. It blooms with white (or pink) trumpet-shape blooms, and can overtake plants readily. It's not an easy weed to control. Any little piece of the root system can resprout a new plant. Your best bet may be to spot treat it with glyphosate (Roundup). Use caution, because the glyphosate is nonselective and will kill or damage other plants it comes into contact with You can dab or brush the herbicide onto the foliage of the undesirable weed, avoiding contact with other plants.