The 1837 edition replaced the indefinite article a with the definite article the. This change was not marked by Joseph Smith in the printer’s manuscript, and it could well be a typo in the 1837 edition since elsewhere “a(n) hundredth part” is the normal phrase in the Book of Mormon, although there is one occurrence of “the hundredth part”:
Summary: Restore “a hundredth part” in the Words of Mormon 1:5 since this phrase represents the earliest reading; “a(n) hundredth part” is considerably more frequent in the text than “the hundredth part”.