The broadside linked by Masato lacks the last (eighth) verse of Lady Dufferin's original. As noted previously in one thread, the lyrics of this popular poem also were printed on a broadside by Andrews, New York. In addition to the posting by Alice in 1997 (linked above by me), the poem is in The Oxford Book of English Verse (at Bartleby.com): Lament Irish Immigrant
I think George Seto posted the Bartleby link previously, but Mudcat is limping today.

The song is also in "Songs of the Newfoundland Outports," vol. 2, pp. 462-463, with two sets of music, one by James and Lucy Heany and the second by Nicolas Keough.
Title given is "I'm Sitting on the Stile, Mary." A note states that the Heanys learned it from "The Family Herald," a discontinued Canadian weekly. How these relate to musical settings by Barker, Dempster, and the Rankins, I do not know.