The Kurt Gödel Constitutional Dictatorship Loophole

I had heard the story of Kurt Gödel discovering an inconsistency in the U.S. Constitution, one that would allow the U.S. to become a dictatorship. Towards the end of his life Oskar Morgenstern, who along with Albert Einstein went to the citizenship hearing for Gödel (all three were Institute for Advanced Study faculty at the time) recounted the events of that citizenship hearing. The exchange went something like this:

And then he turned to Gödel and said, Now, Mr. Gödel, where do you come from?
Gödel: Where I come from? Austria.
The examiner: What kind of government did you have in Austria?
Gödel: It was a republic, but the constitution was such that it finally was changed into a dictatorship.
The examiner: Oh! This is very bad. This could not happen in this country.
Gödel: Oh, yes, I can prove it.

As far as I’m aware, his discovery was never published and was either never revealed or lost through the years. I can’t imagine how three of the greatest minds to ever gather in one location (and they went as far as working in the same institution) could have avoided debating such a topic.

The memorandum from Morgenstern recounting the event (found here) is linked below and mirrored for posterity:
Morgenstern on Gödel citizenship [pdf] | original [pdf]