Here are some more:
Pavane, Op. 50 by Gabriel Fauré
The Aquarium from Carnival of the Animals by Camille Saint-Saëns.
A few by Sidney Bechet:
“Do You Realize” is the obvious Flaming Lips song, but I’ll nominate “All We Have is Now.”
Cracker (and their predecessor Camper Van Bethoven) have some surprisingly good ones, including Big Dipper and All Her Favorite Fruit.
The orchestra version of All Her Favorite Fruit is almost hard for me to listen to, it’s so damn beautiful.
Mine in no order:
Anything by Paul Robeson but this is my fave - Summertime http://www.youtube.com/watch?v=7j5CieXP7L8&feature=related
From Oliver - Who will buy? http://www.youtube.com/watch?v=JqiuKXHk6tM
Russian Red Army Choir - Song of the Volga Boatmen http://www.youtube.com/watch?v=8WD0WVL-HjE&feature=related
Ivan Rebroff
http://www.youtube.com/watch?v=Ep3h-2r_q98&feature=related
An Aussie song to balance! Hunters and Collectors - Throw your arms around me. http://www.youtube.com/watch?v=LJ4Lbn_t5dk
Something for the teenagers - Lady GaGa - Paparazzi Acoustic Version"]Lady GaGa - Paparazzi Acoustic Version http://www.youtube.com/watch?v=l3R3KqrJAI4
Came in to mention this melody. A few more:
Fantasia on Greensleaves by Ralph Vaughn Williams
Nimrod by Edward Elgar.
A lot of old folk songs are nice, too. Scarborough Fair, The Wild Mountain Thyme, etc.
Germany Before the War- Randy Newman. There’s a cover by Peter Mulvey on ‘Ten Thousand Mornings’ that is disturbingly beautiful.
Fish and Bird- Tom Waits. In fact, almost all of ‘Alice’ has that feel.
Well if you mention Blue Bayou, you have to mention Roy Orbison’s version. He got there first.
Hey, speaking of Roy Orbison, how about Crying?
Vaughn Williams - Fantasia on a Theme by Thomas Tallis
Carol of the Birds - Robert Shaw Chorale (especially at 1:20 where only the men sing)
Carol of the Birds - Pablo Casals
Tomas Luis de Victoria - O Magnum Mysterium
Poulenc - O Magnum Mysterium
Bruckner - Virga Jesse Floruit
I’ve heard Orbisons’ version, his voice just isn’t All That AFAIC.
Chicane - Saltwater Vocals by Máire Brennan of Clannad
Part of it is context, but I love Non nobis domine from Henry V: http://www.youtube.com/watch?v=Z1GDRx-F1C0
I want it sung at my funeral. Can’t get more achingly beautiful than that.
Ahhhh yes, Chicane. 
One of the best electronic acts ever. That whole album is so great.
I love Rebekah del Rio’s Spanish version (Llorando - featured on the Mullholand Drive soundtrack)
Although I had enjoyed Buddy Holly’s upbeat rockers for quite some time, it wasn’t until after his untimely death that I felt the ache that these three songs bring:
Raining In My Heart - Buddy Holly
Buddy Holly - It Doesn’t Matter Anymore 45 rpm!
I never even heard this next one until after his death:
Bob Dylan, Visions of Johanna, from the Albert Hall concert
This is one of my favourite songs. Perfect Day by Miriam Stockley.
“True Love Ways” almost made my list.
I can’t stand Holly’s version of “It Doesn’t Matter Any More”- complete disconnect between lyrics and music. Linda’s version is far better, gorgeously sad and beautiful.
.
Wha? His voice is only one of the most beautiful voices in the history of recorded music, let alone the history of rock and roll.
But I guess different strokes for different folks.
I’m partial to the French version. Edith Piaf or Tina Arena, take your pick.
I was searching through this thread to mention that very song. A lot of his work could be in this list.