A half-assed gardener's conundrum.

Our humble little backyard garden has brought me great joy so far this summer. Considering the space amounted to a bramble-filled wasteland before we took it over, and considering that the three of us have approximately five minutes of horticultural experience combined, I'm pretty satisfied with the way things are going. We've got massive sunflowers, taller than I am. We've got plucky little tomato plants, yielding the most delicious little buggers I've had in some time. We've got daisies and basil and the makings of some pretty cute marigolds.
We've also got a rapidly-increasing thatch of vines, lush with spade-shaped leaves and purple flowers. At first, I paid it no mind. Then I started to like it--it looked good, and covered up an otherwise-empty wall of the garden. Then one day last week I noticed it's expanded. A lot. To the point that it's taken over most of our garden and some of our neighbour's.
I consulted my father, who knows about things like this, and he gave some saddening news. Turns out it's a very invasive weed, and it might be choking out all other life in our plot.
So now I'm torn. One the one hand, I don't want my pretty other plants to suffer, and nothing rubs salt in the wound of dead tomatoes quite like parental "I told you so!"s. But on the other...the vine is so pretty! I mean, so what if it's a weed. It can't help it. And I am nothing if not an equal-opportunity green thumb.
So...should I tear the poor little (okay--big) guy up? Or should I let it incite a sort of Darwinian battle in the garden?
Yeah. These are the things that I think about.