Listen to the recording for rhythm and strumming. Lanegan kind of speaks some of the 
words and holds things for as long as he wants, so you can follow his style or add your own.

VERSE 1:
G              F               C
The heart will pound until the breath is gone
G               F               C
You know what I love so, put me on
G              F         C
Something has badly gone wrong with me
G           F                   C
Livin's not hard, it's just not easy
F
Always keepin' the dogs off

CHORUS:
C             D      C             D
Down like the rain, down like the rain
C                    D             G
Comin' down like the rain, comin' down
G    C        D              G
Oh I wanna be there, by your side
            C
Cause this feels like dyin' baby
G              D                  G
Let me stay around, let me stay around

VERSE 2 (same as Verse 1):
You take my heart, and my breath is gone
You know what I love so, put me on
Feel a little light-headed, you know I'd have to be
Believin' ain't hard, it's just not easy
Stayin' blind to the obvious

CHORUS (same as before):
Down like the rain, down like the rain
Comin' down like the rain, comin' down
Oh I wanna be there, by your side
Cause this feels like dyin' baby
Let me stay around, let me stay around

Repeat Chorus, then play through Chorus chords as many times as you like until the end. 
There's a cool solo on the recording, but they're also decent chords to just improvise over.

This is my first set of chords, hope you enjoy!