This offer is for those with the Kindle Wi-Fi with Special Offers or Kindle 3G + Wi-Fi with Special Offers only. Turn on your wireless, click Menu, Sync & Check for New Items, then on View Special Offers after the sync has completed. You should see a new offer:
$10 for a $20 Amazon.com Gift Card
Click on offer, then click on the link on the offer page sign up and get an email with the promotion code. Sign-up for this offer expires on June 21.
You'll get an email (right away) with a link to the special $10 for a $20 Amazon.com Gift Card page. Once you have the promotional code, you have until July 21 to complete your purchase (but, why wait?). Like previous offers, this one requires you to use the full checkout process in order to enter your promotional code. Also, like all Amazon sales that use promotional codes, if you have a gift card balance, you must use it for the payment (if there is not a sufficient balance, then you can pick which credit card or other payment to use).
So, on the special $10 for a $20 Amazon.com Gift Card page, use the Add to Cart button. Then click on Proceed to Checkout, sign in and then enter the promotion code you received via email, where it says Gift Cards & Promotional Codes and click on Apply. Double check your Order total and place the order.
Note that even if you took advantage of the previous $20 for $10 offer, you can still take advantage of this one. In fact, with this offer, you can buy one certificate for each KSO device registered to your account - just click on the ad on each device to receive the individual promotion codes via email.
Once again, the gift card claims it will be mailed, but I expect to see mine in email sometime tomorrow (based on the delivery from the last time this offer hit).