A bit of additional web research has established that subfixer was right: http://www.imdb.com/title/tt0047797/
As I remember it, Above Us the Waves was a pretty good flick. Unfortunately it seems to be available on DVD only in the PAL format, which doesn't work on American DVD players. Its top-billed star, John Mills, also starred in another first-rate British submarine movie called Operation Disaster. I'd really like to see that one again.
Subfixer - over to you for the next question.
In case anybody's curious about the fourth answer I had in mind - I was careful in the original question to use the word "production," rather than "movie." Theodore Bikel, who has had a second career as a folk singer, "created" the role of Kapitan Georg von Trapp, Imperial Austrian Navy, in the original Broadway production of The Sound of Music.
So next time you watch The Enemy Below, think "Edelweiss, Edelweiss...."
Youth, talent, hard work, and enthusiasm are no match for old age and treachery.