There are a lot of cases of songs made of other songs, ranging from novelty record compilations with spoken word sections alternating with pop song answers, to Weird Al Yankovic compilations of pop songs done as polkas to “The America Variations” to Walter/Wendy Carlos’ “Pompous Circumstances”. My favorite is one that I heard out in the deserts of Utah on the radio – it took me a while to realize that the song was made by alternating sections of the theme from Sylvester Stallone’s “Rocky” with sections of the theme from “Rocky and Bullwinkle”. I only heard it once.
There’s an awesome mixture of “Over The Rainbow” and “What a Wonderful World” sing by Israel comeone (he has a LONG Hawiian name that would take me a month to learn to spell and two to learn to pronounce, and I’m too lazy to look it up) I’m really surprised nobody’s mentioned it.
Tricky did a version of “Black Coffee” where the background music was actually built off the piano from Elvis Costello’s “Pills and Soap.” It’s worth a listen - very eerie.
Nobody’s mentioned The Tubes’, * Theme from a Wooly Place* yet. In that one, they play “Theme from a Summer Place” on one speaker while performing “Wooly Bully” in the other chanel.
Though that one’s a bit less than melodious, the whole side (it originally came out in vinyl) of nine songs are linked together through the beat, never stopping betwen songs. And the beginning song is reprised at the end.
If I ever get married, I’m requesting that that “side” gets played at the reception.
Forgot to mention that title’s from their album Love Bomb.
Damn, home brew beer makes it hard to type.
The local NPR show Rewind did a show on mashups a few months ago. At the show’s signoff, they played Eminem’s Without Me vocals over the Morning Edition theme music.
On a tangential note, my sisters and I discovered at an early age that the lyrics to Amazing Grace can be sung perfectly to the tune of Gilligan’s Island.
Amazing Grace, how sweet the sound
that saved a wretch like me;
I once was lost, but now am found
Was blind, but now I see.
Was blind, but now I see.
Etc.
This seems as good a place as any to mention DJ Danger Mouse’s Grey Album: Jay-Z’s Black Album and The Beatles’ White Album together at last.
http://www.rollingstone.com/news/newsarticle.asp?nid=19292
It’s been in heavy rotation on my iPod.
It’s
ISRAEL KAMAKAWIWO’OLE
and I can’t pronounce it either. They just called him ‘IZ’ in Hawaii.
Dude weighed in around 500 lbs. He died. No wonder.
A friend of mine sings Gilligan’s Island to the tune of Stairway to Heaven… He’s a weird guy.
Don’t remember seeing it so far, but the cowboy junkies have a good seque between… er… a song that segues seemlessly into “Blue Moon”. They called it “Blue Moon Revisited (Song for Elvis)”, but I can’t remember what the first son was…
If you’re looking for medley styles check out “Bug Bunny on Broadway” it has “The Rabbit of Seville” which take a whole slew of different musical pieces and connects them all together. I never appreciated the difficulty of transposing a lot of that music until I was reading an article about it.
A cover of Nina Simone’s “I Wish I Knew How It Would Feel To Be Free” by the Lighthouse Family uses U2’s One, as does the Black-Eyed Peas’ “Where Is The Love?”.
Damn! You beat me to it 
Heh heh heh. 
I was quite surprised someone hadn’t beaten me… I was going to start a new Cafe thread for this album (it is very good, y’all), but as it happens there’s already a GD thread about it: http://boards.straightdope.com/sdmb/showthread.php?t=240888
ObNewContent: I’ll add “You got your Herb Alpert in my Public Enemy!” and:
http://evolution-control.com/sounds/gunderphonic/
Funny you should mention that. I have a version of “Stairway to Gilligan’s Isle” by Little Roger and the Goosebumps that dates back to the early 1980’s.
[Hijack] It was heard on the legendary KFAT-FM {“94.5 between the cheeks of your FM dial”) and its subversive country format which flourished in Gilroy, California (the garlic captial of the world) 1977-83. For fans of a mixture of early alt-country, bluegrass, rockabilly, roots music and some truly bizzare novelty songs, check it out on Shoutcast,com. They apparently have days and days worth of tapes they recycle.
And if you’re intrested in the descendent of that noble experiment, check out KPIG-FM in Freedom, California, which even features some of the same DJs. Used to be one of the Net’s greatest freebies, now sadly only available with a monthly fee through RealAudio.
And, nope, I have no connection whatsoever with those two excellent radio stations other than being a fan.
[/Hijack]
More on the point of the OP, I’m reminded of the gimmick in Gulliver’s Travels, the 1939 Max Fleischer full-length feature film that was going to be his studio’s answer to Snow White and catapult him into the ranks of feature film-makers.
Sadly, it was not to be. Though there are a number of nifty sequences, it lacks the gravitas Snow White had, and was not a huge success.
However, the key to the plot was Gulliver washed up on the shores of two neighboring kingdoms of miniature people. Lilliput and Blifuscu. The kings have arranged a royal marriage to bind their kingdoms together but the kingdoms go to war because they can’t agree what song will be played at their wedding: Lilliput’s anthem, or Blefuscu’s. Ultimately, Gulliver solves the problem by suggesting a solution that sound like something from an old Certs commercial (“You’re BOTH right!”).
Why not play them both together? One of them was called “Faithful” and the other called “Forever.” Put 'em together and you get “Faithful Forever.” Awwww.
So here’s the OP as the whole McGuffin for a movie. We’ll leave it up to you whether they count, since the two songs were written specifically to go together by the studio.
If you get a chance to find a video or DVD it would be worth renting.
That was the first one I thought of too. That one worked well.
Unfortunately the second one I thought of was a Merry Melodies/Disney thing where Goofy takes control of an orchestra playing some operative overture–maybe The Barber of Seville–and transforms it into “Arkansas Traveler” . . . and back . . . and then back to “Arkansas Traveler” again. Not a real good example (although it, too, worked well).
Sandra Bernhard did a song called “Maniac Superstar,” which combined Jimi Hendrix’s “Maniac” with Jesus Christ Superstar’s “Everything’s All Right.” Not only does it work, but I am amazed at how good it is.