Dicho, the song was directly inspired by "Hard Times." The tune itself is "Hard Times". The Lester S. Levy Collection of Sheet Music has the sheet music (CLICK HERE). The description is:

Title: [Foster-Hall Reproductions]. The Anniversary And Sunday School Music Book. Number Five. Sorrow Shall Come Again No More.
Composer, Lyricist, Arranger: Words by W.K., from "Choral Hymn Book." Music by S.C. Foster. Arr. by A. Cull.
Publication: New York: Horace Waters, No.333 Broadway, 1859]..
Form of Composition: strophic with chorus
Instrumentation: piano and voice
First Line: What to me are earth's pleasures, And what its flowing tears?
First Line of Chorus: 'Tis a song from the home of the weary
Subject: Biblical references
Subject: Religion
Subject: Death
Call No.: Box: 070 Item: 009

~Masato