Help with source of poem

[maggie9]

At the end of the BBC dramatisation of LOTR is Bilbo's last song which begins
"Day is ended; dim my eyes
But journey's end before me lies."

There are several sites on the web which gives these lines but none gives a proper reference except to a book called Bilbo's last song illustrated by Pauline Baynes.

It's not in LOTR so is it a poem Tolkien rejected?

I'm no great Tolkien expert so apologies if this is a stupid question.