Final answer:
The new capital of the Roman Empire after the split and the fall of the western half was Constantinople, previously known as Byzantium. It served as the stronghold of the remaining Eastern Roman Empire, also referred to historically as the Byzantine Empire.
Step-by-step explanation:
When the Roman Empire was split and the western half fell, the new capital was called Constantinople. This city was originally known as Byzantium before Emperor Constantine moved the capital there and renamed it in 330 CE. Constantinople served as the powerful fortress controlling the Bosphorus Strait, which connects the Black Sea to the Mediterranean. As the eastern administrative center, the city continued using Greek rather than Latin as its official language. Even after the fall of the Western Roman Empire in 476 CE, the Eastern Roman Empire, also known as the Byzantine Empire (a term created later, as they called themselves Romans), continued to thrive until 1453.