Remember that the name Babylon has both literal and spiritual meaning (see 2 Nephi 23:19–22). As is the case with many of Isaiah’s prophecies, there is a dual fulfillment for this prophecy. Babylon became one of the most glorious cities of the world during the reign of Nebuchadnezzar, around 600B.C. It fell to the Medes and Persians in 539 B.C. and began a long decline. By the time of Christ, Babylon was inhabited only by a few Jews exiled by Roman decree. A hundred years later it was totally desolate and has remained uninhabited to this day.
Babylon is also the name for Satan’s kingdom, or the world (see D&C 1:16). In the great judgments that will immediately precede the Second Coming, spiritual Babylon, also known as the church of the devil or the great whore of the earth (see 1 Nephi 14:10, Revelation 17:1–5), will be destroyed and remain in utter desolation during the Millennium.