A systemic weed killer is the only way to rid yourself of this horror.
Of course you could dig it up, but with the amount you say you have it would be to much like hard work!
If you do use roundup or something similar, consider the residue left in the soil if you are going to grow food crops.
With alot of patience,you could cover the area with a black polythene sheet for a season or two which will stop it from photosynthesizing and eventually kill it off that way, but that's only if you're prepared to wait.
Good luck.