The 1840 edition added the indefinite article an before ancient. This addition is probably an error rather than due to conscious editing on Joseph Smith’s part. Perhaps because ancient begins with an, the 1840 typesetter mistakenly repeated the an, thus adding the indefinite article.
Of course, either expression (with or without the an) is acceptable for English. In fact, there is a possibility that the original text read “of an ancient date” and that Oliver Cowdery accidentally dropped the an when writing down the text for the original manuscript. 𝓞 itself most probably read without the an since both the 1830 edition and the printer’s manuscript read without the an. Moreover, there is one other occurrence in the text of “of ancient date” (but none of “of an ancient date”):
This example thus confirms the earliest reading in 3 Nephi 3:9, “of ancient date”.
Summary: In accord with the earliest textual sources, accept the expression “of ancient date” without the indefinite article an (in both Mosiah 8:13 and 3 Nephi 3:9).