Well, yes, but not as popular history has it. The Western Empire crumbled, and the management of Rome itself was taken over by "barbarians", (who spoke Latin and were pretty good administrators). The economy actually improved a little, due to lower taxes and less corruption, until Islam exploded out of Arabia. That trashed the economy of the entire Mediterranean, leading quickly into what Enlightenment historians dubbed The Dark Ages. Dark to them because so little was written down for them to study, owing to the loss of Egyptian papyrus.
Read Henri Pirenne's "Mohammad and Charlemagne" and also Emmett Scott's "Mohammad and Charlemagne Revisited" which adds the findings of modern archaeology to Pirenne's original thesis. Both still in print.
BTW Gibbon, whose gigantic "Fall of the Roman Empire" influences some historians even today, was a rabid anti-Christian bigot. Keep that in mind.
Dammit - you summarized it so much better than I did.
and excellent references