Is the root in your or your neighbour's garden? If it is in your garden have the vine cut back to the ground and then either treat the stump with chemicals or hire a grinder to remove. If the root is in your neighbour's garden you will have to keep pruning back or talk to them about the problem and ask them to deal with.

Was just about to write a virtually identical answer to Moongrower's.

Great minds Inverglen - lol!