HeLlAnGeL

I agree, Mad World by Gary Jules is overly depressing.

Heaven (911 Remix) by DJ Sammy is not so much depressing, but so damn sad it'll make you bawl till you can't bawl anymore.

Other very depressing songs:
Nothing Compares 2 U by Sineod O'Connor
The Christmas Shoes by Newsong
In The Air Tonight by Phil Collins
Asleep by The Smith's
Hide and Seek by Imogen Heap (strange song, but very beautiful)
I'm Sorry by Glassjaw
Konstantine by Something Corporate (You have GOTTA get this song)
Bluebird Of Happiness by Mojave 3
Wonderwall by Ryan Adams (The tune of this is really really depressing)
Forgotten Hopes by Anathema
Rainy Days by Guster
Stan by Eminem (The words are disturbing)

These songs aren't as depressing, but they're really good and they make me want to kill myself for some reason:
Walking By by Something Corporate
Only Ashes by Something Corporate
Good News by Something Corporate:

"And then she looked at me to scream,
'my castles are falling'
But I can't look into the street,
without everything changing

I wanna read good new, good news
I wanna be innocent again
I wanna read good news, good news
But nothing good is happening"