To get a trial offer to Amazon Prime , put a bunch of items into your Amazon.com shopping cart, select one or two day shipping, and on the final checkout screen you should see the following text:
[Your name], you have been selected to receive a FREE four-month trial of Amazon Prime. If you sign up then you can get unlimited One-Day Shipping for $3.99 per item on over a million items sold by Amazon.com while your trial lasts.
» Learn more & sign up  (link added)
The free Amazon.com Prime trial lasts four months after which they will charge you usual $79 annual fee. They even promise to send you two reminder emails! For my thoughts on Amazon Prime, if you pay for it, read this post .
No Purchase Is Necessary!
Update: You can get the free offer by clicking this link:
And to prevent it from automatically upgrading you from the trial (free) to the actual ($79/yr), go to Your Account, scroll down and click Manage My Subscriptions, and then click the “Do Not Upgrade” button. I’d still double check after four months to make sure you aren’t charged.