Albeit 3,300 years ago. The New Scientist asks, Were Cursed Rams the First Biological Weapon?:

Ancient written texts from the Middle East may reveal that the use of biological weapons dates back more than 3300 years, according to a new review.

The historical documents hint that the Hittites – whose empire stretched from modern-day Turkey to northern Syria – sent diseased rams to their enemies to weaken them with tularemia, a devastating bacterial infection that remains a potential bioterror threat even today, says the review.

Experts caution that more evidence is needed to firmly establish that the Hittites intended to spread disease using the animals. But they add that if this proves true, it might represent the earliest known use of biological warfare.

Tularemia, also known as rabbit fever, can pass from animals such as rabbits and sheep to humans through various routes, most commonly through insects such as ticks that hop between species. The bacterium responsible for tularemia, Francisella tularensis, causes symptoms ranging from skin ulcers to respiratory failure.

Interesting bunch, the Hittites. When they weren't busy with their robust bio-weapons program, they developed one of the first constitutional monarchies. In 1550 BC, they successfully sacked Babylon, located in what is now southern Iraq--possibly to prevent the Iraqis Babylonians from developing WMDs of their own.