How do you get rid of honeysuckle vines......

I know there will be those of you who cring at this but I really need to kill off some of my honeysuckle vines. They've taken over a flower bed and every year they try to strangle some of my trees ...... How do you kill these things..... I've spent two weekends trimming away at them. I know they'll just end up coming back from the roots I'm leaving behind....... I need a solution that will not harm my mimosa and redbud trees that are growing in the middle of all this honeysuckle mess and kill all the honeysuckle............ Any ideas ? Any suggestions ? Anything would look better than the mess I currently have.....

Cut it down to the ground and cover it with black pastic for a few weeks :)

Thanks moonlighting........ I was thinking it might be less work in the long run to just leave it covered and make it a rock garden LOL but I'd like to salvage it for a flower bed if possible......