Chapin (1814-1880) wrote the poem when he was still a student at the Bennington Seminary, with the title "Bury Me Not in the Deep, Deep Sea."
He was still in his teens.
The poem is quoted in full in his biography in the section "Schooldays."
"Life of Edwin H. Chapin, D. D.," 1882, Sumner Ellis, D. D., Copyright Universalist Publishing House; John Wilson & Son.

http://www.archive.org/stream/lifeofedwinchap00elli

Mention is made of publication in the Southern Literary Messenger and its copying in other periodicals, but no publication dates are given.
His biographer characterizes him as an orator, poet, wit, mimic, ventriloquist, a singer of songs, and a story teller.