Killing weed/bush intertwined with lilac bush

[Babe flowers]

Hello,
I hope someone can help me. I just moved to a new house that has a big lilac bush. There is a strange type of bush or weed that is intertwined with the lilac bush. It has long sharp thorns and the inside core is yellow. Last year I cut it back as far as I could hoping that it would not grow back but it is. Luckily all I have to do is nick the bark to tell what it lilac and what's the thorny beast! Is there a way I can kill it without harming the lilac bush?

[tomc]

Cut the stem low down on the vine. Like at ground level, low down.

[Babe flowers]

I cut them really low to the ground even dug into the dirt a little and it's still growing back. I thought I heard about painting something on the tops of the stems after cutting them down. Have you heard about anything like that?