Shrek--- Who's singing "Hallelujah"?

In the movie Shrek, there is a beautiful rendition of Leonard Cohen’s “Hallelujah” being sung over scenes where Shrek and Princess Fiona are separated and miserable.

According to the CD soundtrack, it was Rufus Wainwright (sp?), but it sounded like John Cale singing in the film.

So who’s singing “Hallelujah”?

The end credits do list John Cale as the singer. They may not have gotten clearance to include his version in the CD, though, which would account for the seeming discrepancy.