I used some Roundup, as per instructions, and the topgrowth died back after about a fortnight. The roots however, are still white and alive, and ruddy difficult to dig out. (some are dead and blackened, but they are just as tough to dig out).
I'm not convinced that weedkiller is the be all and end all ... there's still a lotta digging to be done!
On my mum's plot, we covered the weeds with carpet and fence panels - after 6 months (winter) the soil is beautiful to dig, there are virtually no weeds remaining, and the ones that are struggling to survive (dock, bindweed, couch) get dug out with little resistance. If you have the time, this is the way to go.

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~ What do I think of Western civilisation? I think it would be a very good idea ~ Gandhi