Atmega328 powered by a 3.7 volt battery?

Sure - research fuses, internal oscillator, 8 MHz.
I've never done it, but I have read instances where others have, so it is possible.

Here's bunch of info on,
http://wolfpaulus.com/journal/embedded/arduino3
likely more than you will be able to use.

Do yourself a big favor & get a 2nd arduino - then you can keep your deumilanove intact for programming & development, and download into something like promini for embedding into your project.