No, that was not an actual copy of the Constitution behind President Obama as he spoke today.
A spokeswoman for the National Archives confirms that just like in the movie National Treasure, the document on display today was a facsimile. The klieg lights in the room were way too bright and harsh for the Constitution, which is usually seen just by dim candle-lights, as seen here.
So the facsimiles were brought out for this event.. The impact of President Obama speaking in the echo-filled chamber with the words "THE CONSTITUTION OF THE UNITED STATES OF AMERCA" etched in marble behind him were not accidental.