Cowboy Junkies doing Lou Reed's Sweet Jane:
[youtube=425,350]http://www.youtube.com/watch?v=BHRFZFmEq9o[/youtube]
For some reason everyone doing Dylan put this song in my head.
That was farking cool. Pretty true to the original yet they somehow make it their own. Never heard it before for sure but I have it saved now, lol.
This next one is a bit iffy because I don't know if it's a cover or not. 
I remember some hub bub about it when the song came out and I know the symphany part is from the Rolling Stones (at least I think I know that) but the song may very well be The Verve. Just don't know. (??)
The Verve - Bitter Sweet Symphony
[youtube=425,350]http://www.youtube.com/watch?v=V-Po8uJeoUw[/youtube]
Wow! I just read the Wiki page about that song. I remember when it happened because Rush ended up talking about it on his show but I didn't realize how complicated the issue was. 1997! Wow, how time flies.
Fine. Make me look it up myself.

Pretty cool stuff. Guess I can't call it a cover really. Cool history though. (to everyone else, look it up your damn selves, LOL)
To make up for my previous lack of research, here is an authintic cover. (of Dylan btw, heh heh)
Norah Jones 'I"ll be your baby tonight'[youtube=425,350]http://www.youtube.com/watch?v=1OJ99iOFTRo[/youtube]