If you have a paid Prime membership (the free student ones don’t work, the $39 student ones will), you can borrow one book per month for free from the Kindle Lending Library.

I’ve identified a couple catches.  This only works if you have a Kindle (the Kindle apps for iOS / Mac / etc. won’t work as far as I know).  Also, Amazon has made it a little difficult to browse the books that are available for free.  However, I found a post describing a workaround for the second problem.  Seems to work great.

  1. Go to www.amazon.com .
  2. Select “Books” in the drop-down menu for the main search box.
  3. Leaving the search box empty, click on the “Go” button. 
  4. Click “Kindle Edition.”
  5. Scroll down on the left hand side and click the “Prime Eligible” filter.
  6. Pick a book and get reading.