So interesting how things morph over the years and tellings. Not sure if this is already well documented, but none of the words that I learned in the 70's are here. Also learned it at Liahona girls camp in No. Cal, and the midi tune posted is similar but quite different in places. We also sang is as a round, which is very pretty.

Here is how we sang it. -
There's a web like a spiders web,
made of silver light and shadow
Hung by the moon
in my room at night.
IT's a web made to catch your dreams
hold them fast till I awaken
as if to tell me
my dreams are alright.

Then - a tag after
There's a web
A spider's web
Hung by the moon in my room at night
it's a web
a spider's web
as if to tell me my dreams are alright.

The other person sings this - then they switch.

Down in the valley
there is a meadow
down by the old oak tret
Down in the meadow
there is a river
where my dreams flow free.