After singing the first verse for many years to my kids at night, I chanced upon the full words to this lullaby fairly recently, in an old book on Astronomy for children, Among the Stars by Agnes Giberne, 1885.

Twinkle, twinkle, little star,
How I wonder what you are!
Up above the world so high,
Like a diamond in the sky.

When the blazing sun is gone,
When he nothing shines upon,
The you show your little light,
Twinkle, twinkle, all the night,

Then the traveller in the dark,
Thanks you for your tiny spark,
He could not tell which way to go;
if you did not twinkle so.

In the dark blue sky you keep,
And often through my curtains peep;
For you never shut your eye,
Till the sun is in the sky.

As your bright and tiny spark,
Lights the traveller in the dark,
Though I know not what you are,
Twinkle, Twinkle, little star.

I especially like the verse about walking by starlight; this is one of the joys of having moved back into the countryside (or virtually so; despite neighbours insecurity lights). I have actually walked by starlight in Moycullen; quite a change from the city lights of Dublin.

Further investigation on the web shows it was written in 1809. Should be safely out of copyright, unlike "Happy Birthday".