This is the version my parents sang to me, and I in turn sang to my children... and still do on occasion, though they're 9 and 13...

Go to sleep my baby,
Go to sleep my dear,
Great big moon is shining,
Stars will soon appear.
Close your little eyes,
Now you must not peep,
Fot it's time for little piccaninnies
To go to sleep...
Time for little boys/girls to go to sleep.

(Tune seems to be very similar to that in the clip that Mary in KY linked to above.)