I've taken this from what I've heard of Glen Hansard 
and Marketa Irglova's version and 
posted on the site. I'm fairly sure it's right but 
there are bits where you can vary if 
wish.

You can play the F as a barre chord (133211) 
or you can play an F6 variation (x30210) 
roll your first finger onto the high F in 
the shape of F6 (x30211). When he strums F 
into the chorus he seems to hold the low E 
with his thumb. The strum for it is DDD UDUDUDU

There are a couple of variants, the intro and verse 
can be played either of two ways


Intro:
  C                   F                 C                  F
E----------0------------------1----0-------------0--------------------------|
B-1---3----------3----1---3-------------1---3---------3----1---1---1---1----|
G---0---0----0-----0----2---2---2----2-----0---0----0----0----2---2---2---2-|
D---------------------------------------------------------------------------|
A-3-------------------3-----------------3------------------3----------------|
E---------------------------------------------------------------------------|

Verse 1:
  C                   F                 C                  F
E----------0------------------1----0-------------0--------------------------|
B-1---3----------3----1---3-------------1---3---------3----1---1---1---1----|
G---0---0----0-----0----2---2---2----2-----0---0----0----0----2---2---2---2-|
D---------------------------------------------------------------------------|
A-3-------------------3-----------------3------------------3----------------|
E---------------------------------------------------------------------------|
 I don't know you     but I want you    all the more for that
  C                   F                 C                  F
E----------0------------------1----0-------------0--------------------------|
B-1---3----------3----1---3-------------1---3---------3----1---1---1---1----|
G---0---0----0-----0----2---2---2----2-----0---0----0----0----2---2---2---2-|
D---------------------------------------------------------------------------|
A-3-------------------3-----------------3------------------3----------------|
E---------------------------------------------------------------------------|
Words fall through me and always fool me and I can't react

Or

Intro:
  C                   F                 C                  F
E----------0------------------1----0-------------0--------------------------|
B-1---3----------3----1---3-------------1---3---------3----1---1---1---1----|
G---0---0----0-----0----0---0---0----0-----0---0----0----0----0---0---0---0-|
D---------------------------------------------------------------------------|
A-3-------------------------------------3-----------------------------------|
E---------------------1------------------------------------1----------------|

Verse 1:
  C                   F                 C                  F
E----------0------------------1----0-------------0--------------------------|
B-1---3----------3----1---3-------------1---3---------3----1---1---1---1----|
G---0---0----0-----0----0---0---0----0-----0---0----0----0----0---0---0---0-|
D---------------------------------------------------------------------------|
A-3-------------------------------------3-----------------------------------|
E---------------------1------------------------------------1----------------|
 I don't know you     but I want you    all the more for that
  C                   F                 C                  F
E----------0------------------1----0-------------0--------------------------|
B-1---3----------3----1---3-------------1---3---------3----1---1---1---1----|
G---0---0----0-----0----0---0---0----0-----0---0----0----0----0---0---0---0-|
D---------------------------------------------------------------------------|
A-3-------------------------------------3-----------------------------------|
E---------------------1------------------------------------1----------------|
Words fall through me and always fool me and I can't react

You can always fill the gaps in the bridge with additional notes
if you think it's necessary.

Bridge:
  Am           G        F            G                 Am
E--------0--------------------1----------------------------|
B--------------3---1--0--------------3----1----0-----------|
G------2---2--------------2------2-----------------------2-|
D----------------------------------------------------------|
A--0---------------------------------------------------0---|
E--------------3--------1------------3---------------------|
   Games that ne-  ver a-mount to    more than they're meant

Again the F can be played a number of ways. 
As shown, or (1x3211) or (1x3011). Either way, I like to hammer on
the E string as I strum the F chord. I don't know what it's called but
he also does something like a trill for violin as he begins to play the F,
watch or listen to the song to see what I mean.

         G                F(strum)
E--0----------------------1----------------|
B--------3----1----0------1----------------|
G----2--------------------2----------------|
D-------------------------3----------------|
A-------------------------3----------------|
E--------3----------------1----------------|
   will play  them-selves out.

In the chorus the first F in every line seems to be  (1x3211)
holding the E with the thumb, and the second is (1x3013),
though you can just play plain old F if you like.

Chorus:

C                 F                 Am                   F
Take this sinking boat and point it home we've still got time
C                   F                 Am                    F
Raise your hopefull voice you had the choice you've made it known

Verse 2:

  C                   F                 C                  F
E----------0------------------1----0-------------0--------------------------|
B-1---3----------3----1---3-------------1---3---------3----1---1---1---1----|
G---0---0----0-----0----2---2---2----2-----0---0----0----0----2---2---2---2-|
D---------------------------------------------------------------------------|
A-3-------------------3-----------------3------------------3----------------|
E---------------------------------------------------------------------------|
  Fa- lling slow-ly   eyes that know me and I   can't go back

  C                   F                 C                  F
E----------0------------------1----0-------------0--------------------------|
B-1---3----------3----1---3-------------1---3---------3----1---1---1---1----|
G---0---0----0-----0----2---2---2----2-----0---0----0----0----2---2---2---2-|
D---------------------------------------------------------------------------|
A-3-------------------3-----------------3------------------3----------------|
E---------------------------------------------------------------------------|
  Moods that take me  and e-  rase me   and I'm painted black

Bridge:

Am           G        F            G                 Am
E--------0--------------------1----------------------------|
B--------------3---1--0--------------3----1----0-----------|
G------2---2--------------2------2-----------------------2-|
D----------------------------------------------------------|
A--0---------------------------------------------------0---|
E--------------3--------1------------3---------------------|
  You have     suffered enough and   warred with yourself

         G                F(strum)
E--0----------------------1----------------|
B--------3----1----0------1----------------|
G----2--------------------2----------------|
D-------------------------3----------------|
A-------------------------3----------------|
E--------3----------------1----------------|
   it's  time that you    wo-on

Chorus:

C                 F                 Am                   F
Take this sinking boat and point it home we've still got time
C                   F                 Am                    F
Raise your hopefull voice you had the choice you've made it known
C              F               Am             F
Falling slowly sing your me-lo-dy I'll sing a-long

he then strums the chorus a couple of times, strums F a few times 
and returns from the verse to end