You can copy-paste between two instances of Eagle. You can also copy the board, close Eagle completely, re-open Eagle, start a new board, and paste what you copied back in. I just did both methods with Eagle 6.4.0.
One thing to remember is that you have to close the schematic window (if it's open) before you paste in the board window. Otherwise Eagle will not allow you to do that - it has no way of adding to do a backwards update to the schematic.