This is the exact way Damon Albarn plays Melancholy Hill in this acoustic version that 
was on BBC 1 :http://www.youtube.com/watch?v=7iODSZL7XkQ

Damon uses his thumb over the top of the neck holding down the low E string. 
Alternatively you can bar all the strings using your index finger on the same fret that his thumb 
is on in every instance of the song.

Here is the tablature that I made (my first tab!). The pattern repeats throughout the 
whole song.

  D
e|---------------------------------------------------|
B|---------------------------------------------------|
G|---11----11----11----11----11----11----11----11----|
D|------12----------12----------12----------12-------|
A|------------12----------------------12-------------|
E|-10---------------------10-------------------------|
  A
e|---------------------------------------------------|
B|---------------------------------------------------|
G|----6-----6-----6-----6-----6-----6-----6-----6----|
D|-------7-----------7-----------7-----------7-------|
A|-------------7-----------------------7-------------|
E|-5-----------------------5-------------------------|
  Bm
e|---------------------------------------------------|
B|---------------------------------------------------|
G|----7-----7-----7-----7-----7-----7-----7-----7----|
D|-------9-----------9-----------9-----------9-------|
A|-------------9-----------------------9-------------|
E|-7-----------------------7-------------------------|
  G                        A
e|---------------------------------------------------|
B|---------------------------------------------------|
G|----4-----4-----4-----4-----6-----6-----6-----6----|
D|-------5-----------3-----------7-----------7-------|
A|-------------5-----------------------7-------------|
E|-3-----------------------5-------------------------|

  G                        A
e|---------------------------------------------------|
B|---------------------------------------------------|
G|----4-----4-----4-----4-----6-----6-----6-----6----|
D|-------5-----------3-----------7-----------7-------|
A|-------------5-----------------------7-------------|
E|-3-----------------------5-------------------------|