It sounds like a good idea Kimmy but you will have to think about how deep you plan these beds to be and also there is the worms to think about! They won't be able to get into your beds and I think they are an important part of keeping your soil in good repair.
I know what you mean about keeping this weed under control but I wonder if there isn't a better alternative? (Hopefully someone will have some good suggestions!)

I think you will regret not digging most of it out before making your beds on top.

I think you may mean Bind Weed, I use a weed killer that is like a big lipstick and rub it on as many of the leaves I can, it draws the weed killer down to the roots and I have found it works but you have to keep doing it as new comes up, it does help but I dont think we will ever completly get rid of it. Good Luck.

i would use the membrane on the top and use a mulch over that as worms are importent.they aerate the soil .turn it over for you .maybe if you put a lot of rubble at the bottem and take out as much of the weed as drc726 sais .thats what ive done with the same problem.if you look on my blogs somewear you will find my trianguler raised bed.i hope this has helped take care buy for now .

If it is Bind Weed, then I think a membrane is a waste of time. Bind Weed can grow a single root 3 meters a year, and it grows like spaghetti, roots everywhere. From my fence it goes under a 12ft shed, like long white ghostly creepers, out the other side , into the grass and then through the lawn.

The good news, it is really easy to dig up if the soil is damp. Just make sure you pick up every piece, as it will re-root from a 5mm long piece. Then just weed kill every little leaf you see coming in the following years.

If your neighbour has it near their fence, then you will get it as well. The idiot who owned this house, thought it was a flower and let it infest half the garden for 10 years. After 1 season I have it pretty much under control.