This is what I found on the official Damien Rice message board:

Basic chords for Older Chests are C then a quick F G x2

then Am followed by quick B C

then F followed by a quick G

Thats the intro/ chorus, & I think the verse is the same. Again some inventiveness is required for picking, but there's the basics

And someone else wrote:

Basic chords are C then a quick F G x2
then Am followed by quick B C
then F followed by a quick G

Thats the intro/ chorus. I think the verse is the same.