This is a wrinkle that people often don't get about the ident attribute: in CVS or Subversion, it would be legitimate to have a VERSION file containing nothing but $Id$ as a way of tracking that the right push had happened; if you did that with git, you'd get nothing useful. If you run the script below (or just read it and the output that follows), you can see why.
(Please direct comments on this to https://plus.google.com/+TreyHarris/posts/KRgNG3zDk9x.)
Setting it up in Git is relatively simple. You create (or edit) the file .gitattributes and add a line telling Git to replace $Id$ when found in files that match certain filenames: