If you send things via Priority Mail (say in a thin Priority Mail envelope) with Delivery Confirmation, it can be any thickness. But other mailpieces that are NOT sent via Priority Mail AND use DELIVERY CONFIRMATION must be over 3/4" thick, and if you send a mailpiece with Delivery Confirmation that is 3/4" or less (or under 3/4"), they charge you the Priority Mail rate.

A lot of people do --and should-- add bubble wrap or Styrofoam worms to the inside of the mailpiece to make it thick enough.

http://www.antiquegamblingchips.com/USPostageApril2011.htm

Robert