I built my boxes last year over an old, fallow, very weedy garden bed. I used cardboard underneath and filled with five inches of Mel's mix. I mulched between the boxes with wood chips. I had very few weeds, so I am a believer. I had one weedy nemesis which was my only weed. I don't know what it was, but it grew from a rhizome up to to cardboard layer, then it would run along for many inches or even feet until it found a hole or gap in the cardboard, and boom, up it came!

At the end of the season, I lifted the mulch and some of the boxes to try to vanquish the unbeatable rhizome. Most of the cardboard was totally gone, or barely there.

My garden, for the most part, grew very well in five inches of soil in boxes made with 2x6 lumber.

If I were to do it again, I would use a second layer of cardboard or maybe some newspaper layers. The vegetables don't seem to care to go much deeper than six inches, and the weeds can get through an incomplete cardboard layer if they are really determined.

I would remove the old soil put down cardboard and newspaper and then put at least 8"of compost and peat moss on top. Although most plants besdies root crops will work in only 6" of soil more will encourge roots to go down in stead of out. If the roots go out near the surface of the soil they are more likely to be disturbed by you planting close by and they will dry out quicker. This really only holds for the large vine crops and the root crops. But it is best to have a foot of soil IMHO.