We don't have firm evidence for an exact end of the cult, but it was sometime after 240 CE/AD. The city of Assur actually experienced a bit of a Renaissance in the late 2nd Century CE and the primary temple of the god Assur was rebuilt. That lasted until the destruction of the city by the Sassanid Shah Shapur I.

I describe more about the post Assyrian Empire history of the Assyrians in this thread