Here's what I do - it takes time, but way less energy than digging.

Stockpile heavy corrugated cardboard (appliance and furniture boxes are great, but the grocery store is a good source also) and an organic mulch that you won't mind looking at for a growing season. (My personal choice is natural wood chips or shavings, but chopped leaves will work also. I avoid dyed mulch since it's hard to judge what has been camouflaged by the dye.)

Mow the area really short or clip or hoe it to ground level.

Cover the area with the cardboard with overlaps - no gaps at all, and cover the edges with something to weigh it down - bricks, heavy chunks of wood, etc.

Cover the cardboard with at least 4 inches of mulch and water well to settle it. The combination of cardboard and mulch will smother your weeds and then will over time decompose to give you soil of wonderful quality.

Wait a year to plant - and watch like a hawk during that year, pulling any tiny sprig of green that might emerge. Resist the temptation to plant now since gaps in the mulch will allow your persistent weeds to regrow. During the year you can plan how you will plant it. To get ideas, visit public gardens, wander around neighborhoods with gardens you like, go on garden tours, and visit nurseries over the course of the season, not just in spring. If you really need to plant, you can put out pots of annuals or even of perennials & shrubs hardy to 2 zones colder than you are (if the pots are frost-proof.) If you are planning vines for your fence, you can shop for or create your trellises. If you don't have an edging that discourages the lawn from entering your beds, you can do that also, either a cut edge or an edging that is buried to keep the grass in the lawn.

So if you do this now, you can plant the bed next spring. Just pull aside the mulch, plant, and tuck back in without piling mulch against the stems. Since you will have chive and sweet peas seeds that may have survived, watch for them and pull immediately. Or spread corn gluten around the plants you put in - it's a natural discourager of seed sprouting, but won't affect your newly planted plants.

I'd love to see an after photo when you've got it planted!

Sweet peas are annuals and should be dead with the first frost, I don't see any in that photo unless we don't call the same flower a sweet pea. Those yellow flowers have a much different foliage than the sweet peas I know.

Anyway, what to do depends on how fast you want to do it. You've already been given advice on how to do it the slow but natural way. The fast way is to use Round Up.

Round Up is very safe if used according to directions. It does not have any residual effect in the soil. You can plant right away after it kills the weeds and you prep the soil.

Perennial sweet peas are impossible almost to get rid of. Roundup didn't work on mine, but maybe I need it stronger. However, I agree with above post, I don't think the yellow flowers are sweet peas. Maybe linaria butter & eggs. Mine pull up easily, but that bed is very loose soil. Round up does work on linaria.

Per nhbabs's advice, I inherited a weed-overgrown neglected garden. Last year in early June I did what she recommended, but my area was very large so I used rolls of 10-year heavy-duty landscaping tarp and 6" of mulch. Cardboard would be so much better as it decomposes rather than just sits there. The areas around existing plants did get cardboard so I could piecemeal around the existing plants. This year I swept away the mulch and pulled up the tarp. All those pernicious terrible deep-rooted weeds were either dead or terribly debilitated. I could easily dig in the soil and remove any tenacious ones that showed any signs of life.

During last summer I did all kinds of planning as to what I would put in the beds. Then I found out about winter sowing and had many many people generously donate all kinds of seeds to me as a new gardener. A few of my plans were fulfilled, but I had so many wonderful new seeds that my plans basically had to start over. But, trust me, I'm NOT complaining in any way. Having those plans wiped has introduced me to so many wonderful plants I had never considered. They are now all seedlings growing in my practically weed-free garden. What a nice change from last year!

If you have any plants in the area you do want to keep, you may need to dig them up, shake the dirt off the roots, and then remove all the weed roots tangled in the plant roots. I had to do that to a few of the existing plants. Do this after the weeds are dead next year.

Yes, it's kind of hard to tell from the photo (taken a year ago) but there are wild sweet pea plants mixed in with all that mess. You can see some of it poking out over all the other plants the left side.. and there's a little pink flower right at the edge on the left. I wish I had taken a better photo recently because it was much, much, much worse than what you see above but I ripped out what I could of it before I posted this thread and am waiting for it to grow back so I could trace more of it to the roots. I'm going to try nhbabs' slow method with cardboard as I have so many other garden projects I have to work on that I'm fine with having it go for a year. If I see some of it next year I'll probably use some Round-Up, but I'm hoping that it'll be like deanna's case where I can just easily dig up the roots.

Thanks for all your advice everyone!!

There is hope yet :) Happy gardening :)

Ah, I see them now. If you wait til a good rain & go out to dig, you can probably get them fairly easily, just dig deep. Or maybe the Roundup will work for you better than it has for me. The main thing is to keep at it. It will get tired of trying after a while.