Here is the fingerpicking pattern, which is played during the whole song:
  G                                   Em
E|-----------------|-----------------|-----------------|-----------------|
B|-------3---------|-------3---------|-------0---------|-------0---------|
G|-----------0---0-|-----------0---0-|-----------0---0-|-----------0---0-|
D|----0--------0---|----0--------0---|-----2-------2---|-----2-------2---|
A|-----------------|-----------------|-----------------|-----------------|
E|-3---------------|-3---------------|-0---------------|-0---------------|

  C                 D                 G
E|-----------------|-----------------|-----------------|-----------------|
B|-------0---------|-------3---------|-------3---------|-------3---------|
G|-----------0---0-|-----2-----2---0-|-----------0---0-|-----------0---0-|
D|-----2-------2---|-0-----------0---|-----0-------0---|-----0-------0---|
A|-3---------------|-----------------|-----------------|-----------------|
E|-----------------|-----------------|-3---------------|-3---------------|

If you want to play the chords, instead of the fingerpicking you can play them as showed here.
I found these chords like they described here sounded the best to the original recording.
You can also play them without the pinky finger on the high G.

    G    Em   C    D
E|-3----3----3----2--|
B|-3----0----0----3--|
G|-0----0----0----2--|
D|-0----2----2----0--|
A|-2----2----3-------|
E|-3----0------------|


[Intro]
G    Em   C  D   N.C.

[Verse 1]
           G
When the night has come
Em
  And the land is dark
          C          D               G   N.C
And the moon is the only light we'll see
         G
No, I won't be afraid,
      Em
No, I won't be afraid
         C            D          G
Just as long as you stand, stand by me.

[Chorus]
                      G
So, darling, darling, stand by me,
    Em
Oh, stand by me.
    C     D            G
Oh, stand, stand by me,
Stand by me. 

[Verse 2]
          G
If the sky that we look upon
Em
  Should tumble and fall
          C            D              G
Or the mountain should crumble to the sea,
          G
I won't cry, I won't cry,
      Em
No, I won't shed a tear
         C            D           G
Just as long as you stand, stand by me.

[Chorus]
                       G
And, darling, darling, stand by me,
    Em
Oh, stand by me.
    C         D            G
Oh, stand now, stand by me,
Stand by me.

[Instrumental]
G    Em   C  D  G    x2

[Chorus]
    N.C               G
So, darling, darling, stand by me,
    Em
Oh, stand by me.
      C              D
Oh, stand now, stand by me,
      G
Stand by me.

[Ending]
                                       G                Em
Whenever you're in trouble won't you stand by me, oh, stand by me
      C       D            Em
Oh, stand now, stand by me, ohhhhhoouuuu
         C            D           G
Just as long as you stand, stand by me.