It's actually too late this season to do the solarization. Here in Ohio, days are already getting shorter and temperatures moderating. For the solarization to be very effective, especially on really tough stuff like ivy and poison ivy, the plastic needs to be on for at least 6 wks in height of summer (which of course means you basically can't use that area that whole growing season). Best is 2 layers, black plastic on bottom with clear plastic over it, that way you get the weed suppression of the black and the heating of the clear. And Bobber, when you are trying to get rid of ground covers like ivy, it really doesn't matter if the soil doesn't heat real deeply. Ivy doesn't root very deeply, it's all pretty much in the top inches.
But, yeah I like the cardboard method. Ivy being very tough, I would lay down multiple layers of cardboard. Wet it thoroughly to aid in breaking down and maybe walk on it a bit to reduce air pockets under it. Then you can lay down several inches of topsoil and plant into that. The cardboard smothers the stuff under it and then rots away, leaving a nice planting bed.