I had serious problems with chickweed after using huge quantities of manure collected from a farmer who is an neighbour of mine. Two years of careful hand weeding has cleared the lot and I now have a garden that is just about chickweed free. If your weed barrier is like a thin membrane then it probably won't do the job it's designed to do. I've discovered that the only thing that works is the really heavy duty material used commercially; it's made to stand up to having tractors driven over it.

Use of a pre-emergent will eliminate this pest. Dachtal (if you use such things), or apparently corn gluten also acts as a pre-emergent seed killer. It will probably take more than one application, but this treatment will eventually help you.