Oh man, I feel your pain! Bishop weed is notoriously tough to get rid of. I tried once, by removing the top 6" of soil, but it grew back anyway. I was glad to move away from it (incidentally, the BW didn't chase me away. LOL!) If faced with that plant again, I would remove the desirable plants and smother the whole area for a year, or 18 months if starting in the fall. Then uncover a small test area and make sure it's safe that following summer before assuming it's dead.

Another approach could be to smother a new garden spot now, to move desirable plants to in the spring. Then smother the BW spot for as long as you need. I don't know how much space you have available, or what might sound better to you.

I wouldn't use plastic though, just corrugated cardboard under at least
6" of organic matter. Might as well improve the soil while smothering,
not kill all of the microbes.

I would rinse the roots of the desirable keepers as each one comes out of the old bed. The smallest pieces of root from BW can stay alive, keep growing into a whole new invasion. That should also prevent the moving of tiny purslane (or any other kinds of) seeds into the new area, if that's an option for you.

Totally agree, seeds don't go well in mulch, if they go at all. You might want to remove most of the mulch (if it hasn't yet decomposed) from a newly smothered area in which you want to have sprouting seeds. If it's been there through a winter, the cardboard should be gone. But, when removing mulch used for smother, be ready for other seeds that were in the soil to then be able to sprout. The first year of any new spot is the toughest, as the "bank" of existing seeds exhausts itself. If you can get through the first year of pulling stuff before it drops new seeds, sprouts of those plants should be dramatically less the following years.

A hard border around beds can help keep grass out, and mowing with the chute aiming away from beds. Timbers, bricks, that kind of thing.

And what do I do with very established shrubs, like my pinky wink hydrangea, etc? can they withstand the heat of the neighboring black plastic, or is that a good reason to just use cardboard?

same query elsewhere

http://forums.gardenweb.com/discussions/3292648/i-need-to-do-a-total-renovation-of-my-flower-beds-weed-invasion?n=2

I spent around 30-40 hours removing soil & sifting through what was left trying to get rid of BW that time (mentioned above.) The spot was about 100 sq ft. That totally failed. A smother will succeed if left alone long enough. BW is tough, but not a strong enough plant to break through a heavy layer of mulch over well-overlapping cardboard, like tree suckers might. Everything unwanted has to be dead before it's disturbed.

I would pull up the shrubs too, so a solid, unbroken smother can be used over the whole area. I'm really anal about unwanted plants, and can't enjoy the good ones if something is invading, wouldn't want a Hydrangea that has BW around it.

Furniture & appliance stores are great places to seek large pieces of cardboard. If you call and ask about it, they should be able to help you out, put some aside for you & help you load it up. Remove any tape or staples. Mom'n'pop stores are probably your best bet, ask to speak to the warehouse manager.

If you're able to add a pic of your problem spot, that could inspire folks to have more specific input to offer.

Tiffany, if you go to my post in " weeds" in Houzz, you can see the photos of my garden. (I am having some computer issues at the moment). My hydrangea is going to be too big to move. I guess i am really looking for an answer that doesn't exist, because I know all the usual answers are going to involve crazy-hard work and/or losing some good stuff! (My desirable invasive, like obedient plant and monarda, are being invaded by the undesirable ones!)

TY! I saw your pic - and don't blame you for continuing to solicit advice. It's BIG, and some of the shrubs are massive, moving just one would take a while. I couldn't see BW in particular, but if it's creeping through that whole area - ugh!

Brainstorming past actual experiences:

I might try is string-trimming everything unwanted, as often as possible, like every few days, but most unwanted stuff isn't going to give up from just that. But it would at least prevent seed-dropping, and thriving weeds while you ponder &/or work on another area.

If I was losing the battle still, I might consider using some kind of 'cide, the way people describe doing with a sponge brush, so only the amount needed to coat particular leaves is used. I might try straight vinegar first, but wouldn't be willing to lose such a big, beautiful garden to being otherwise unable to evict an unwanted plant, if that didn't work. Painting individual leaves sounds like a lot of work too.

Or I might consider moving everything but the shrubs out & trying to smother under them, paint-brushing unwanted growth right at the base of the shrub trunks, if/as needed.

Sympathy & encouragement, for whatever you try!