Yes, Morwen, it's a composed song that has become somewhat of a standard in the folk and bluegrass genres. See this message - apparently the lyrics were published in 1854, written by Sarah T. Bolton. There have been various melodies tied to these lyrics.

I'm still puzzled about the lyrics in the Digital Tradition and in Gale Huntington's Songs the Whalemen Sang. Huntington's lyrics reportedly came from the 1853 journal of the Lexington. This page which quotes a letter from The Lexington, has an additional verse about the stepmother singing the mother's song. Anybody want to go to the whaling museum in Nantucket and check the Lexington journal?
-Joe-