Cut back yellow jessamine vine severely and dig the roots from the ground. Remove any pieces of stem that have rooted into the ground. This is the best way to kill the vine, but not necessarily the easiest if you have a vine that has grown wild for some time.
Fertilize heavily and often with a nitrogen-based fertilizer, such as a 23-0-0 made for lawns. The fertilizer will burn the roots and kill the plant eventually. This method works best with younger vines.
Cut the vine back severely in the spring and paint all cut parts with full-strength brush killer. Spray new growth with brush killer as soon as it appears. This works well on vines with large trunks that would be difficult to remove by digging.