I would probably say a new date which is 1204.
By the time Rome fell, the capital of the Roman empire actually had changed to Constantinople. In it's later existence of the late 2nd century onwards, Rome focused even more heavily on it's four quarters (it had been separated into four quarters for a while, which caused all sorts of problems at times with Western vs. Eastern Emperors, the catalogue of discourses and wars between the various factions...etc) which were primarily the North and South and East and West, making it the Northwest, Southwest, North East, and South East quarters. By the time Rome Fell, we could say it was more of a focus of the Eastern portion of the Empire, with the Western portion (of which Rome was a part of) having fallen in repute and economics for at least a century.
The real heart and what was considered the empire at that point was in the East. That empire continued for several more centuries.
Rome, the city fell, but ONLY Europe (and specifically Europe that was constrained by the Roman Catholic Church which had a vested interest from the inheritance of Charlemagne) actually considered that meaning that the Roman Empire fell...or that Rome fell. This was for legitimacy in their part, as they were trying to show legitimacy of rule and right...a legitimacy rejected by others.
However, when trouble hits, as it did, sometimes lines blur. In the late 11th century, Rome was in trouble. They had incursions from tribes that were threatening it's capital and so it asked it's brothers in Christianity for aid. Ironically, rather than just aid the slowly fading royalty of the Byzantines (who were the Romans), they went on crusade (now known as the first crusade) and went on to cause trouble all along the way until they left on the Southern borders and went on to conquer Jerusalem.
This same idea of crusade eventually led to the 4th crusade, where the final remnants of Rome, it's rulers and aristocracy were overthrown and replaced by Roman Catholic counterparts and the ideas that the Roman Catholic Church expoused in claiming that this was the Byzantine...not the Roman...empire.
It was at that point I think that the old Roman empire truly finally died in it's death throes (death throes that had been ongoing for several centuries at that point) and a short lived Roman Catholic nation existed.
If one doesn't accept that, I'll take the later date where it falls from Christian hands to Islamic hands, but I don't think the fall of the city of Rome actually was the Fall of the Roman empire as has been popular amongst nations heavily influenced by the Roman Catholics in their noted history books during the 18th - 20th centuries. It is an idea that I think is finally starting to fade away as other historians and ways of looking at history are finally being seen by the forefront of history today and a wider acceptance of history beyond that of Western European dialogue is finally surfacing.
