Who set that exchange rate?
The Free Market.
Before the end of the Civil War, the Confederate dollar was worth less than wallpaper --- basically worthless as a medium of exchange. Kind of like the Zimbabwe dollar today. People in the South didn't want to accept it either because they knew it was worthless.
That is why so little of it remains today. People used it as kindling by that point in time just as we would use a day old newspaper.
If you have some Confederate Currency today however, you could get some serious US Greenbacks for it from collectors just because it is so rare. And it's only rare because it was so worthless.