Fall of Ancient Babylon

Below is a compilation of OT Scriptures discussing the destruction of Ancient Babylon

2 Chronicles 36:17-23
This passage is obviously not a prophecy, but does show the history involved with the following stages:
1. Neb/Babylon destroy Jerusalem (17-19)
2. The 'land' was desolate for 70 years (20-21)
3. Cyrus/Persia allows them to return (22-23)
In this passage the exile is said to have lasted 'until the kingdom of Persia came to power'

So that is what happened historically

As far as prophecy goes, we'd want to look first at the end of the 70 year prophecy of Jeremiah

Jeremiah 25:12-14
In this passage Jeremiah prophesies that:
1. Babylon will be destroyed when the 70 years is over (12-13)
2. Babylonians will be enslaved by 'many nations' (14)
If your bible has footnotes, you may be directed toward Jeremiah 50-51 which sheds more light on the destruction of Babylon

Jeremiah 50:9 (also 50:41)
This passage predicts an "alliance of great nations from the land of the north" will bring down Babylon

Jeremiah 51:27-28
This passage lists some of the members of that alliance: Ararat, Minni, Ashkenaz, & the Medes

Next, we would turn to Daniel 5 (the writing on the wall) which includes a mixture of prophecy AND history.

Daniel 5:26-31
In this passage, Daniel is apparently brought out of retirement so that the Babylonian King Belshazzar can understand the mysterious writing that was written on the wall during his party. The writing was not hard to read, it was hard to interpret

Mene, mene, tekel, parsin

The last word, parsin, was interpreted by Daniel to indicate: "Your kingdom is divided and given to the Medes and Persians" (28).

This prophecy became history that very night (30)

So now Daniel is living under the rule of the Medes and Persians. In the first year of that rule, Daniel (who apparently didn't return to Israel with the others, probably b/c of age) is reading Jeremiah 25 (see Daniel 9:2). He recognizes that the 70 years has ended, but still feels the need to repent for his sins and the sins of his people (9:20).

This is the context of the much disputed 9:20-27. The 70 years had ended, now a 490 year prophecy was given to replace it. But I suppose I'm getting off topic!

One last reference, perhaps the most important on the subject at hand. Isaiah, about 200 years prior to the event, predicted the defeat of Babylon:

Isaiah 44:28-45:2
This passage prophecies that God will use Cyrus (amazingly identified by NAME, 200 years before he was famous) to decree the rebuilding of Jerusalem. It even tells how God would bring him to power: "I will break down gates of bronze and cut through bars of iron" (2).

This is interesting, historically, because we know from history that Babylon (considered unbreakable in that day) was defeated when Cyrus's army broke through a gate blocking underground passage via the river which flowed into the city.

Conclusion
Hope one of these passages was what you were looking for. Some of the difficulties involved are the fact that Scripture speaks of 'many nations' defeating Babylon AND of Darius defeating Babylon AND of Cyrus defeating Babylon.

I think from these Scriptures it is best to guess that Cyrus was the leader of Persia which was the most powerful section of a broad alliance. Darius may have been the lead general of the army that defeated Babylon.

But there is little debate, that I am aware of, that this happened at the end of 70 years of captivity under Babylon. Cyrus, upon obtaining the remaining Jews, issued a decree which ended Jeremiah 25's prophecy and started Daniel 9:20-27's.