How do I get rid of this stuff!?

I'm not a gardener, and don't really have any great interest in working
in the yard. A number of years ago, my mother gave me a few sprigs of
mint since I liked that in my iced tea. I stupidly planted the mint in a
backyard flower bed (raised with cement walls). The mint took over the
bed, which is probably 15' long (length of a small townhouse backyard)
and swallowed several rose bushes that had been planted by the previous
owners. Now the entire bed is comprised entirely of very tall, wild
mint! My lawn guy cuts it down several times a year, but is there any
way to eliminate it? I have some roundup that I used on my poison ivy -
would that work? There is nothing else in the area that I care about
preserving (heck, there's nothing else in the area!). The only caveat is
that my neighbors have a small garden abutting the mint bed. Since there
is a cement divider, it hasn't taken over there... yet.

Thanks for any help.

Lee
(and yes, it is a real address, just not one I check as often)