Cedar mats really well and is my favorite. That over the paper should work.
Just a thought - when we needed to obliterate the thistle/bindweed in my yard, we solarized using black plastic and rocks to hold the plastic down. I have *never* seen a trace of either weed in my yard since (well, except for a few thistle babies that come up along the fence line from the neighbors' yard!). This does sterilize the soil but if you wait a bit and work in some compost, leaf mold, etc. you'll be able to plant in it next season. We just used black plastic from Home Depot and weighed down the edges with rocks without burying them. We left it on all through July and August.
You may not need to go to that extreme but I am grateful every year as I de-bindweed my front yard that we nuked those suckers in the back! :)
http://www.organicgardening.com/learn-and-grow/soil-solarization
http://www.hortmag.com/weekly-tips/qa/solarizing-soil
Good luck!

Thanks, I think I'll pick up a load of cedar mulch today and see how that goes.

I carefully spot applied roundup on all of the bindweed in that bed this spring and it has now died off. I am not sure I can keep it that way - unfortunately there is field bindweed all around in the surrounding empty area next to the garden (neighbor's place). So it invades on a regular basis seeking moisture...

The thought of having bindweed eradicated for good would be a dream come true. :)

Thx,
Michelle