Is there a simple way to do this?
You can always use Robomongo. As of v0.8.3 there is a tool that can do this by right-clicking on the collection and selecting "Copy Collection to Database"
For details, see http://blog.robomongo.org/whats-new-in-robomongo-0-8-3/
This feature was removed in 0.8.5 due to its buggy nature so you will have to use 0.8.3 or 0.8.4 if you want to try it out.