Hi Blonde Heather,
If your honeysuckle is the cream and white fragrant one, it's the invasive Japanese honeysuckle aka Lonicera japonica 'Halliana' aka Hall's honeysuckle aka purple honeysuckle aka Lonicera japonica 'purpurea'.* Lots of names for the same awful vine.* Since you say it's stopped blooming, you may not know which one you have.* If the stems are a purple color you should suspect it is.
I do know that people have cut the Japanese honeysuckle to the ground and it resprouts.* If it's the invasive one you have, I would like to suggest you dig it out, paint your arbor and replant a native.* Hmm, I just noticed how you spelled 'arbour'.* You may be in Canada or the UK.* If so, I would still suggest you plant a native from your area.* If you let me know where you are located I will help you choose.* There are many varieties of honeysuckle that are native to North America that rebloom all season long from spring to hard frosts in winter, offer berries to the birds that aren't invasive and some are even semi-evergreen in the warmer zones.
When weeding, the best way to make sure you are removing a weed and not a valuable plant is to pull on it. If it comes out of the ground easily, it is a valuable plant.