Here's your proof from the Amazon forums
(for Kindle 2, I don't know why it would be different for the DX).
Posted on Feb 26, 2009 4:20 PM PST
Amazon Kindle Customer Service says:
Good Evening. This is the Official Amazon.com Customer Service post for 2/26/2009:
Kindle 2 Battery
We learned from the first generation Kindle that changing the battery was rarely needed, so Kindle 2's battery is not user replaceable. The battery is covered under the initial one year warranty as well as the extended warranty offered on Amazon.com. Once outside the one or two year warranty period you can get it replaced for a $59 fee (plus $8 shipping & handling). If you encounter any kind of power or charging issues during the life of your Kindle 2, you can contact Kindle Support for assistance.
Also the support page
for the battery states:
CTIA 1725 Battery Safety Statements
The rechargeable battery in your Kindle should be replaced only by an Amazon authorized service provider. To prevent damage to your Kindle or injury to yourself or others:
Edit: I see you found your information while I was typing.