Although Amazon does not give a direct number for Kindle support, as of 2015 the number for Amazon customer support is available on Amazon's FAQ page. Kindle users are also able to log into their accounts and enter a phone number to receive a call from Amazon's Kindle support team.
In order to access Amazon's Kindle support line, it is necessary to log into an Amazon.com account. In the far right column there is a Kindle help home link under the heading "Need Support?" The Kindle customer service page has a column entitled "Quick Solutions" with a "contact us" box. Click contact us, pick the device in question, select an issue and then select to be contacted by phone. Enter the number and select either "call now" or in 5 minutes.