You just have to give yourself a good day and just weed and till, weed and till, and weed and till some more. But after a day, the job should be done. Of course, you'll still have to pull up weeds throughout the growing season, but that's just how people have done it forever. Nothing like organic gardening to let you appreciate why people wanted to invent herbicides and pesticides in the first place.
Replacement will not be worth it, especially if you have, as you say, really rich soil full of good critters. But 150 sq feet isn't a lot. I have a 500+ sq foot garden and I was able to get most of the weeds in one weekend.