I wasn't satisfied with the two existing versions of this, so I decided to give it a try.
Chords are based on Ella's quite well known 1933 version of this wonderful standard 
featuring the vocal quartet "Ink Spots".

Key: Bb

Verse (00:09)

Bbmaj7   Bdim   Cm7    F9
Say it's only a paper moon
Cm7      F9    Bbmaj7       (F9)
hanging over a cardboard sea
Bb7                Eb6   Edim
but it wouldn't be make-believe
   F9              Bbmaj7   F9
if you believed in me

same for next Verse

Yes, its only a canvas sky
hanging over a muslin tree
but it would be make believe
if you believed in me

Chorus (00:40)

Eb6     Edim   B/F
Without your love
       Cm7   F7      Cmaj7
it's a honky-tonk parade
Eb6      Edim  B/F  G7
without your love
       Cm7                  F7
it's a melody played in the penny arcade

Verse (0:56)
same as first verse

It's a barnum and bailey world
just as phoney as it can be
but it wouldn't be make-believe
if you believed in me

(Ink Spots)

Say it's only a paper moon
hanging over a cardboard sea
but it wouldnt be make-believe
if you believed in me

Yes it's only a canvas sky
hanging over a muslin tree
but it wouldn't be make-believe
if you believed in me

Chorus (1:43)

as before,
repeat twice


Possible chord voicings:
You can always make up your own, depending on what effect you're aiming for & what 
instruments you're plaing with. These are just some suggestions.

Bbmaj7: 6x776x or x1323x
Bdim:   7x776x or x2323x
Cm7:    x3534x  , x3131x or 8x888x
F9:     1x121x or x8788x
Bb7:    x1313x or 6x676x
Eb6:    x6554x or x6558x
Edim:   x7878x
B/F:    1x333x or x8876x
G7:     3x343x or xx5767

For those interested:
The Melody

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


e|------------------3----------|--------------------------------|
B|---------6--6---7----4-------|----8---6-----------------------|
G|-----------------------5-----|--------------------------------|o
D|---3--3--------------------8-|-7-----------7--8--------(8)----|o repeat
A|-----------------------------|--------------------------------|
E|-----------------------------|--------------------------------|


e|-----------------------------|--------------------------------------------|
B|---------------8--6-6--------|-----------------6--6-6--6-------8--8-8--8--|
G|----5--6--7--8---------7-7---|----5--6--7--6-7------------6-7-------------|
D|-8-------------------------8-|-8------------------------------------------|
A|-----------------------------|--------------------------------------------|
E|-----------------------------|--------------------------------------------|


Hope this could help you.