They are using completely different technology. Please work with your library to determine the best course of getting your checked out book on your Kindle.
Moving this topic to Adobe Digital Editions.
Once it shows up inside ADE
Click and drag on of your ebooks from the library then drop it over the KINDLE icon.
What Chesterr, just wrote will never work. Kindles are absolutely not supported by ADE.
Sorry, Chester. This won't work. Digital Editions does not support
Kindles, as Amazon has their own unique version of software.