How big of a bed are we talking about here?
Either way, I'd still recommend removing the sod first. There are lots of hand-tool sod cutters out there that will get the job done fairly quickly for a fraction of the cost of a gas-powered sod-cutter.
There are two type of readily available 'hand' type sod-cutters...
Full hand...
and kick-type...
The kick type makes it a lot easier to get uniform strips of sod rolled up that can actually be placed elsewhere much the same way a gas-powered sod-cutter will. The full-hand one, if kept sharp, just gets the job done without so much concern for being neat about it.
Barring all that, if you do plan to just lay down cardboard and compost, it really does come down to what exactly is growing in the area you wish to convert. It could be as quick a a week or two (not likely) to much longer for full kill.
