This is a very interesting question. The sovereign nations issue their own currencies and are taken for granted by many people, but few consider whether this is the economically optimal option.
All countries would go bankrupt and a huge war will start.
Governments can run because they can print money (or through a system of banks).
If they can't print money anymore. roads, hospitals, schools, police, army would all be gone. resulting in a state of anarchy. I am not saying this is good or bad but it will not be peace.
The largest part of that single currency will end up with the elite while governments have no control at all (because they don't exist anymore).
Unless someone finds an algorithm that gives governments the capability to create that single currency to the amount that is inline with their countries economical performance, trusting any other country. basically the question would be, would you trust the leader of North Korea with printing your money?