That tune is from American IV: The Man Comes Around.

Cash has been doing a lot of covers in the past decade with Rick Rubin. They are all pretty good, but different enough to not be able to call them "better", like Rusty Cage (by Soundgarden) and One (U2). On the IV album, he also covers Desparado (Eagles), I'm So Lonesome I Could Cry (Hank Williams), Bridge Over Troubled Water, and In My Life (Beatles).