On the LP "The Lonesome Boatman" the Furey's were singing "Dance around the Spinning Wheel" which I also have seen titled as "The Spinning Wheel" or as "Dance with me". In one of my sources it is attributed to D. Murphy, in another to "trad". Here's what I hear the Furey's singing:

DANCE AROUND THE SPINNING WHEEL

1. Oh then dance around the spinning wheel,
grab your partners for a reel,
dance around and around the spinning wheel,
and I'd love to dance with Peg O'Neil.

Chorus: Oh, then dance with me, oh my charming one,
with your golden, golden slippers on.

2. I'll buy you silken hose to wear,
I'll buy you ribbons all for your hair.
The sun and moon for you I'll steal
if you'll dance with me, my Peg O'Neil.
Chorus

3. Hands across and make the ring,
all together now let us sing.
We're going to dance the Sligo reel,
and I'm going to dance with Peg O'Neil.
Chorus: So then...and it's dance, dance, dance.

Wolfgang