What was the name of the eastern portion of Rome that continued intact for 1000 years after the Fall of Rome?

The part of Rome that continued to exist intact for such a
long time after the "fall" of Rome was commonly known as the Byzantine
Empire.

By the time that Rome "fell," the part of the Roman
Empire that was actually ruled from Rome was called the Western Roman Empire.  There was
another part of the Empire, known as the Eastern Roman Empire, that was ruled from
Constantinople, which was also called Byzantium.  This is the city that is in what is
now Turkey and is now called Istanbul

After the Western
Roman Empire "fell" in 476 AD, the Eastern Roman Empire continued to exist.  It came to
be called the Byzantine Empire.