Nature Boy by David Bowie. I also enjoy Celine Dion's version. :)
Um, I don't know if this what you're looking for, but try 'I Am Strecthed On Your Grave' by Dead Can Dance. Really haunting...
Oh! And 'The Highway Man' as sung by Loreena McKennit (sp?)
Oh, oh, OH, You and I by Jeff Buckley!! Actually... anything by Jeff Buckley. Everybody Here Wants You and Opened Once are two of my other favourites. They're on his Sketches For My Sweetheart The Drunk album. Then, on his album Grace we have his cover of Hallelujah, which is just incredible no matter who does it, Dream Brother, Lilac Wine... aw, hell, just get some Jeff Buckley!
(Fan? Me? Ha!)
Err... I'm on a real jazz fix now, actually. Billie Holiday. Anything by her. She's amazing. As is Ella Fitzgerald. I just adore their voices.
The Cure's "Longsong," Sinéad's "Stretched On Your Grave," Sara Craig's "Mr. Right," Depeche Mode's "Enjoy the Silence" ... that comes to mind. I'll post more.