Over the cardboard layer, add a thick layer of organic matter such as homemade compost from your compost mile, well-rotted manure, leaf mulch, straw, or grass clippings. Firm it down and tread it in.
Then, with a bit of patience the soil will have become so rich that the weeds can be pulled out with ease. Bear in mind that this process can take 6-12 months, so have plenty of patience.
Basically, due to the soil not being disturbed, it doesn’t bring weed seeds to the surface to germinate.