Keep the weeds mowed real short so they do not go to seed and produce more weeds. Using a pre-emergent on the area at the right time will help too.

Cover it with black plastic held down with bricks for about a month....until all beneath is dead. No need for preen, the heat from the sun will kill the weeds in the top inch of soil.
Then in a s,all part of it plant something like hosta...or blue carpet juniper....some sort of sturdy plant to hold the soil. Do that a little at a time and keep the rest covered with plastic....as weeds emerge in the new plantings....zap them with a spray weed killer.
Or kist mow the weeds short and sprinkle grass seed and keep watered....repeat and once a year hit it with Weed and Feed....keep adding seed and eventually you will have lawn there....but you have to keep mowing.

Soil solarization is best done with clear plastic (think car windows) after wetting down the soil.

tj

Black plastic works by light exclusion, rather than heat. It's not the same as solarisation. In my cool climate weeds just grow better under clear plastic. They think they’re in a lovely warm, moist greenhouse. Black plastic works fine.

Same here. You can grow amazing weeds under clear plastic. While we can get high daytime highs, we almost always cool down at night.hn

Another method is to layer newspaper &/or cardboard, after mowing the weeds as short as possible (if you have a string trimmer, use that to perhaps cut lower than your mower can), then cover the paper layers with mulch or compost. Then you will have a good medium for planting a ground cover or grass as the paper gets turned into soil by worms.

I personally would spray Iron-X, or some other chelated iron weed killer (they are non-toxic to animals and grass), before the mowing - give the weeds a couple of days to absorb it. I would probably also be trying to loosen the soil with a garden fork or spade, and then pulling the largest weeds out. A multipronged approach.

I have used black plastic also to kill weeds and grass in areas that I wanted to make flower beds, it worked pretty well - but I did scalp the area first.

Nope! The nut grass loved the clear plastic in a North Alabama summer. Also, the cardboard. Both were supposed to kill it and all other weeds. The nutgrass, poke weed, and trumpet vine said thank you very much! I am trying black plastic next. If I can find a black tarp even better.