It is constitutional, but only if you listen very carefully, so much so that it has yet to be noticed.
I have just watched this video of the oath of office administered by Chief Justice John Roberts a few hours ago. Unrealized by those who have watched it, it was a confirmation of an historic change that he initiated on January 20, 2009.
This is a long story, one that I have been directly involved in for over four years, and recently took up again by attempting to publicize the meaning of the change. It is important that this be made widely known, as it is, at the least, a small vindication of a long effort by Mike Newdow and others. I have two articles on this on my personal site, the first that I have been working on for the last week, Presidential Oath of Office-to God or Country At the end of the article is a link to an addendum on legal and media aspects of the suit brought by Newdow that most likely resulted in this new variation of the oath.
I'm still working on this addendum, but have it opened for those interested. For those of us who value protection of our secular society this change of a single word is meaningful.