I had the same thing happen to one of my Weigelas! I wasn't sure it was honeysuckle last year, I just kept cutting it down to the ground (knowing it would come back), but I let it grow and start to bloom this year so I could identify it--it was honeysuckle. Just this past Sunday I got my husband to dig it up (it was rooted right next to the Weigela's roots) & I said a prayer that we haven't disturbed the Weigel's roots, at least not enough to do any damage. I also have groundcover by them, though I have sedum and luckily there was enough room between it and the Weigela so as not to disturb it, though it would be much cheaper to replace sedum than a mature Weigela. Hopefully some good advice will come our way as to this situation...

Vine-X
follow instructions on bottle exactly.
You paint it on the ends of the vine.
Don't be stingy with it.
It works.
Round up doesn't.
The sooner the better. You can buy it online.

If honeysuckle grows wild in your area the birds go for the fruit and then drop the seed in the garden.

Sometimes a coarse mulch such as bark nuggets will help you see the young plants before they establish so you can just weed them out.

If the poison isn't fully effective then you may need to thoroughly chop out the root and burn it because it can restart from mere nubs.