Weed killer

The recent sun and rain have been fantastic for the allotment; everything seems to have leapt to life - including weeds. My last couple of sessions on the plot have involved a never-ending cycle of weeding and sowing seeds. Sadly the fine weather is having a positive effect on the weeds as it is on me. I always hand weed the shoots out as soon as I see them but they keep coming back. I dug out as any of the deep white roots as I could, but it's a bit like digging for Australia - you'll never get there. As an organic gardener, I can't apply chemicals to the plot. But I'm sure my regular attacks with fork and spade will one day eliminate these knotty problems.