This was apparently recorded by Danish TV in Sept. 1967. I won’t link directly to it here, but just search on YouTube. The video is incorrectly labeled as from 1968, so search for “Tom Lehrer 1968” and it should be the first result.
Bonus: at about 25:00, Aristotle’s version of the elements song. 
Ah! I’ll have to track it down. Does the show include “Vatican Rag”? I always dance about cackling maniacally when I hear that.
The songs are almost exactly as they are on the common recordings, and much of the inter-song monologue is likewise verbatim, but every so often there’s a new aside or wry joke, and really, the monologue is at least as funny as the songs, so I enjoyed it.
Also, is it a Danish thing to only rarely react during the songs? It seems to me that an American audience would laugh more during the songs. Maybe it’s a generational thing, not a cultural thing. Beyond that, I think that the Danish audience enjoyed some songs more than others; “I Wanna Go Back To Dixie” didn’t get much reaction, but “Send the Marines” was well received.
EDITL Here’s the link. - YouTube
Wow. Apparently the line in “National Brotherhood Week” where Lehrer notes that “Everybody hate the Jews” was not funny in 1967 Denmark. Probably should have seen that coming.
Surely it’s cultural. Granted that (as he notes at the beginning) Danes tend to be well-taught in English, there are so many American cultural cues in the lyrics–and in the genre references woven into the music–upon which the humor is predicated, that I’m sure they just don’t experience the kind of immediate reaction to the references, semantic wordplay, etc., that Americans do. I image the Danes “get” the propositional dimensions of the humor (cf., the marines and hegemony), but miss a large percentage of things like the dissonance of Gilbert and Sullivan musical references to make political/social satire. So obviously they’re going to appreciate the more straightforward songs more.
Danes do tend to speak good English, and I think these are all fans anyway, but still, I’m betting there’s a lot in the songs that they didn’t get.
I’m about 2/3 of the way through and enjoying it–yay Aristotle Elements!–but the applause drives me bonkers. I used to live there and Danish people always seem to do that with the clapping in unison–I didn’t like it then, and now, having to hear it every 3 minutes makes me grind my teeth and want to stab things.
Lehrer’s attempt at Danish is pretty cute. And really bad, but nobody can pronounce it anyway.
Thanks guys, I love tom lehrer, and look forward to being able to sit and watch this one.
For example, the whole “my home town” trope (both musically and ideologically), is probably something that doesn’t even exist in Denmark, so the primary tension of the song is lost on them. To them, it could it just sound like someone singing about criminals or psychos from his city for no apparent reason.
Slightly off-topic, but yowza, he was a handsome man behind those black-rimmed spectacles. Mmmm. 
What’s he say when he’s speaking Danish?
The first time it’s just a little throw-away line that I can’t remember. The second time, when he quotes “a Danish proverb,” it’s “Life is like a sewer; you get out of it what you put into it.” I have no idea if that’s an actual proverb or not.
The concept of Tom Lehrer intrigues me more than his music does (and I love some of his songs): a guy who gave up a lucrative career, in spite of millions in offers over the years, because he didn’t enjoy touring and didn’t like singing the same songs all the time- hard to believe. (I wonder how many of his students over the years had no idea of his past life.)
I can’t imagine that too many of his students were entirely unaware of his past because that’s the kind of thing that students talk about in the cafeteria, especially if they like a professor. I know he didn’t entirely give up singing; someone around here mentioned that there exists a recording of Lehrer with two of his colleagues from the Physics and Chemistry department singing a parody song to convince freshmen to join their respective departments. I think it was to the tune of “Fugue for Tin Horns” from Guys and Dolls. (I got the horse right here, the name is Paul Revere…)
In 1985, in a youth hostel in Reykjavik, I met a student from UC Santa Barbara who’d had a math class with Tom Lehrer. oooooooooooh.
According to her, it was mostly just something everybody knew about him, but not in a big-deal celebrity way. I guess when a guy is standing up in front of a blackboard three days a week explaining what the determinant of a matrix is and so on, without any singing or comic dialogue in the process, you get used to thinking of him as a math professor rather than as a former avant-garde comic.
One of my sister’s household heirlooms is a letter signed by Professor Lehrer, explaining that he no longer performs publicly, and cannot provide entertainment for her wedding. But he was gratified to have been asked.
I don’t know if that’s entirely true. He took part in a revue in 1998 honoring Cameron Mackintosh, who devised and produced Tom Foolery based on Lehrer’s music. The show was a benefit for the Royal National Institute of the Blind, was hosted by Julie Andrews, and the Queen was in the audience, so tell your sister not to feel too bad.