I've heard this song sung to the air of Fíl a Rún. (At least I remember the chorus was the same as given here)
Is that the tune the Fureys sang? The singer did not remember where he learned the song. In Waltons ''Sing an Irish Song'' book 1 (Dublin, no date given, price 9d) it is attributed to D Murphy (singer Delia Murphy, better known for collecting and adapting songs than for composing them). these are the lyrics as given in the Walton collection:

DANCE WITH ME

Dance round the spinning wheel,
Pick your partners for a reel,
Three times round the spinning wheel,
I'd like to dance with Mary Neil.
Get your partners into line,
Keep in step and mind the time,
Now we'll dance the fairy reel,
And I'm going to dance with Mary Neil.

chorus
Dance with me O charming one,
With the golden slippers on,
Dance with me O charming one,
With the golden slippers on.

I'll buy you silken hose to wear,
I'll buy you ribbons for your hair,
The sun and moon for you I'll steal,
If you'll dance with me, dear Mary Neil,
Hands across and make the ring,
All together let us sing.
Three times around the spinning wheel,
I'm going to dance with Mary Neil.

First to the heel and then to the toe,
Round about and off we go,
Circle to left and round the wheel,
I'm going to dance with Mary Neil,
I'll buy for you a gown of green
Finer than you have ever seen,
You will look grander than a ueen,
If you dance with me, dear Mary Neil.

Swing you partner round about
Now to left and in and out,
And then around the spinning wheel,
I'm going to dance with Mary Neil,
Your gown of red and white and green,
Trips around the spinning wheel,
Your fairy feet I cannot feel,
You dance so lightly, Mary Neil.