The Cartesian Mind-Body problem

On the topic of the embodied mind, I'm also watching Yale's open course on Death, with Professor Shelly Kagan. In lectures 5 and 6, Professor Kagan brings up a very interesting philosophical puzzle first posed by Descartes. It is, in brief, as follows:

"Because I can imagine my mind as completely distinct and operational without my body, the mind and body must be two separate things."

Among us weirdo philosophers, this is the sort of stuff that keeps us up at night. I can see how it would be a somewhat compelling argument, after all, it does seem terribly hard to imagine music without sound waves. For that matter, to take a hint from Yeats, it seems rather difficult to imagine a dance without a dancer, or a dancer without a dance. That in itself is a whole separate philosophical debate, but I hope you can see the essential point of Descartes' argument - that in some way, our imagination seems limited by laws of identity and non-contradiction, that is, if a thing is itself, we can not even imagine it to be existent in part but not in whole.

Kagan goes on to list other illustrations:

Can we imagine a smile without a body (and yes, teeth do count)
Can we imagine a podium without a podium?
Can we imagine anything without that anything itself?

So, if we can imagine the mind without the body, if we can imagine being wholly conscious beings without a body (and I, at least, dream fairly often that I am without a body) then must the mind and body be separate things?

I think I've come up with a solution, but I want to hear anyone's comments first, because I'm interested to see if anyone comes up with a solution, and, hey, suspense is fun, right?