Do you have time? I'd blanket the entire surface with black plastic and wait a month.

Sounds like Euphorbia cyparissias, 'Fen's Ruby', such a beautiful plant but nearly drove me out of my mind until I got rid of it. Solarization doesn't work for everything (I've had bindweed in full sun survive 6 months of solarization); The raised bed with rock surround may allow you to get the temps up high enough to kill; I hope it will work for your Euphorbia.

Any vinegar fans out there?

I've never invested in the horticultural vinegar but the reputation is good.

Sometimes (somebody should say it) the only way to do things right is to tear it all down and start over.

Tom

I would second Debbie's rec for sheet mulching. I have never been a fan of solarization for this area. Unless you can provide full sun for pretty much all day long, solarization more often than not doesn't work all that well here. I've seen weeds growing just fine in the moist zone UNDER the plastic, even black plastic. I have had much better luck with sheet mulching myself.

If this infestation is contained to one bed, I would also be tempted to do a variation on the multi-till method too. Basically this means you deliberately leave the roots to sprout, water to get things sprouting again, and go back in once a month to weed them out again, more or less "tilling" to keep the soil loose and watering repeatedly through the summer to get this stuff (and any other weeds, incidentally) to sprout, weed it out, and repeat as necessary to get rid of it. Unfortunately, sometimes this is the best answer you can get.

In this case I don't think round-up is gonna be much faster of a short cut. A good rake to loosen the soil and physically remove the stems and as much of the roots as will come with them is probably the best control you can have, especially if there is nothing else left in the bed.