correction:
Shule, shule, shule aroon,
Shule go succir, agus shule go kewn,
Shule go dorrus agus ellee lum,
Iss guh djay t(h)oo mavorneen sla[i]wn <-- kind of like slah-een

Siúil, siúil, siúil a rúin         
Siúil go sochair agus siúil go ciúin
Siúil go doras agus éalaigh liom
Is go dté tú mo mhúirnín slán

walk, walk, walk (siúl) my [secret love] (a rún)
walk peacefully and walk quietly
walk to the door and escape with me!

The song is lovely. The "Gone the Rainbow" chorus and verses are a clear mangling or morphing of the song, "siúil a rúin" right down to cry my fill / every tear would turn a mill....