There are more options than you may think with plants. As a start there are 186 photos of shade plants on Goypeadia why not take a look?

on the up side the fact that you haveweeds meansyou can grow shade lovers like furns there instead . just get some weed membrane and put the plants in you want and put a mulch over the area .

How big is the border and why is it so shady? Is it under trees, or just in shade from a fence or building? What part of the country are you in, and which way does it face, this area?

There are quite a few options for ground cover in shade. More if it is damp shade, fewer for dry (under trees for instance).
Euphorbia amygdaloides, epimediums, bluebells, ivy (I like variegated as ground cover), hardy cyclamen, lamium species, pulmonaria, dicentras, aquilegia, solomon's seal, hardy geraniums, anemones, primroses, periwinkle. I found a pretty variegated plant at a plant fair last year -
Disporum sessile variegatum and I'm very pleased with it.

I grow all of these except pulmonaria, which is on my wish list for the next GC trip.