The Constitution Obama was using as a backdrop was a fake (http://blogs.abcnews.com/politicalpunch/2009/05/fascimile-con-1.html). The amount of light used for the press conference would have been damaging to the document. Obama was referring to 'Robert Gates' when he said "Defense Secretary “William” Gates".
Videos embedded below.
Full text of Obama's speech can be found at http://www.msnbc.msn.com/id/30866448.
Full text of Cheney's speech can be found at http://www.msnbc.msn.com/id/30867685