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 (xx0232@1)                  DmWhere (xx0231@1)she walks, no flowers bloom
Cadd9 (x32030@1)   G (320003@1)  DHe's (xx0232@1)the one I see right through
D (xx0232@1)                DmShe's (xx0231@1)the absinthe on my lips
Cadd9 (x32030@1)      G (320003@1)   DThe (xx0232@1)splinter in my fingertip

Chorus:
     F (133211@1)       C (x32010@1)   Dm (xx0231@1)   Dsus2But (xx0230@1)who could do without you?
     F (133211@1)     C (x32010@1)Cadd9 (x32030@1)Asus2And (x02200@1)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 (xx0232@1)             Dm (xx0231@1)        Cadd9 (x32030@1) G (320003@1)   DOh, (xx0232@1)we're a pretty, pretty pair, yes we are
D (xx0232@1)                    Dm (xx0231@1)              Cadd9 (x32030@1) GAll, (320003@1)all the king's horses and all of his men couldn't
        F (133211@1)       C (x32010@1)   Dm (xx0231@1)Dsus2 (xx0230@1)       Tear us apart   Oooo
        F (133211@1)       C (x32010@1)   Cadd9 (x32030@1)  Asus2 (x02200@1)       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