>> So my question is ...... I would like to record the song.  What
should I 
>> buy to record that is decent but doesn't cost the world ??

I have used a standard microcassette recorder with a zoom microphone for
this type of application, with great results.  The quality is not high
fidelity, but it's absolutely good enough for ID purposes, and because
these are so small, you can carry it with you every time you bird.  

Digital is probably more convenient these days (plus would make it
easier to share the sound file with fellow birders to get their input if
you don't know what you're hearing). 

I don't know what the digital variety costs, but I'll bet you can get
one for less than $100 at a Radio Shack or Best Buy.

Good luck!

Miriam Lawrence