Corgan did this a few times on the TFE tour. He did it in DC and it was beautiful, much better than that video. He began by saying someone taught him to sing a prayer and he would like to share. He said he believes in God, but in a God that accepts everyone no matter who they are. That said he was raised Catholic so he was going to sing a catholic prayer because he didn't want to sing some quasi PC prayer to appease everyone. Then he sang Hail Mary. Then he exited the stage. It was pretty awesome.
He also did this the previous year in DC at the poetry reading I believe.