If you can till weekly for about a month you will get most of the grass. I almost quadrupled my garden size this year over last year, and asked this very question last fall. I'm planning to mulch heavily with straw to keep the grass at bay. It seems that scalping the grass as short as possible before tilling helps as well.
My first year I used weed proof fabric to cover lots of area. It worked ok, but won't take much foot traffic before it falls apart.

I think he's saying that he gave his wife the plot he has been using the cardboard on, so now he wants to know how to start a new plot using conventional methods

Not sure on your particular situation, but tilling it several times will be a very good start.

If you have the time, a pre-emergent herbicide can stop just about everything from growing (and when applied at the correct time, will allow you to grow a garden in the area). If you don't have the schedule down, then spraying a post-emergent will be a huge help.

Be sure to till in humus in the future if you have access to the quantity you need. This will make decent soil in to very good soil in a short period of time.