The correct answer is option 'B': 539 BCE.

Explanation:
The Achaemenids were a dynasty of ancient Persia (modern-day Iran) that ruled from approximately 550 BCE to 330 BCE. They were known for their vast empire, which at its height stretched from Egypt to India.

The conquest of Babylon by the Achaemenids is an important event in ancient history, as it marked the end of the Neo-Babylonian Empire and the beginning of Persian rule in the region.

The Achaemenid king who conquered Babylon was Cyrus the Great, the founder of the Achaemenid Empire. Cyrus had already established himself as a powerful ruler by defeating the Median Empire and uniting the various Persian tribes.

The conquest of Babylon by Cyrus the Great can be dated to 539 BCE. Here's a detailed explanation of the events leading up to the conquest:

1. Rise of Cyrus the Great:
- Cyrus the Great became king of Anshan, a small kingdom in Persia, around 559 BCE.
- He gradually expanded his territory by conquering neighboring regions and tribes, including the Medes.

2. Fall of the Neo-Babylonian Empire:
- The Neo-Babylonian Empire, ruled by King Nabonidus, was facing internal problems.
- Nabonidus spent much of his reign outside of Babylon, leaving his son Belshazzar in charge.
- Belshazzar's leadership was ineffective, and the empire was weakened.

3. Cyrus' Campaigns:
- Cyrus saw an opportunity to expand his empire and launched a campaign against the Neo-Babylonian Empire.
- He initially captured the city of Opis, located on the Tigris River, in 539 BCE.
- The Babylonian army, led by Belshazzar, marched out to confront Cyrus but was defeated.

4. Capture of Babylon:
- Cyrus approached Babylon and laid siege to the city.
- The Babylonian defenses were strong, with massive walls and a moat protecting the city.
- However, Cyrus devised a plan to divert the flow of the Euphrates River that ran through the city.
- With the river level lowered, his troops were able to enter Babylon through the riverbed.

5. Cyrus as the Conqueror:
- Cyrus was welcomed as a liberator by the people of Babylon, who were unhappy with the rule of Belshazzar.
- He presented himself as a benevolent ruler and allowed the temples and religious institutions to continue functioning.
- This policy of tolerance towards local customs and religions helped him gain the support of the diverse population of his empire.

In conclusion, the Achaemenids of Iran, under the leadership of Cyrus the Great, conquered Babylon in 539 BCE. This event marked the end of the Neo-Babylonian Empire and the beginning of Persian rule in the region.