I was hoping to find chords to this wonderful song already on the web, but I had no luck 
with my search.
So here are the chords I came up with after listening to the song carefully.
I had some trouble figuring out how to label each chord because some aren't normal or 
common, but I did my best.
To get the strumming down, I recommend trying to play along with the recording. Enjoy!

Verse:
D                    Dm
Where she walks, no flowers bloom
Cadd9     G    D
He's the one I see right through
D                  Dm
She's the absinthe on my lips
Cadd9        G     D
The splinter in my fingertip

Chorus:
     F         C     Dm	    Dsus2
But who could do without you?
     F	      C  Cadd9  Asus2
And who could do without you?

Verse:
She's the sea I'm sinkin' in
He's the ink under my skin
Sometimes I can't tell where I am
Where I leave off and he begins

Chorus:
But who could do without you?
And who could do without you?

Bridge:
D               Dm	         Cadd9   G     D
Oh, we're a pretty, pretty pair, yes we are
D                      Dm                Cadd9   G
All, all the king's horses and all of his men couldn't
        F         C     Dm  Dsus2
        Tear us apart   Oooo
        F         C     Cadd9    Asus2
        Aaaaah	        Ooooo

Verse:
Dancing with a ball and chain
Through it all we still remain
Butterflies around the flame
Till ashes, ashes, we fade away