I’m not sure that we can point to one moment in time and say ‘this is when the dole stopped.’ I assume that you are talking about the city of Rome, but there were other doles in other cities, notably Constantinople but also other cities had doles or subsidised food (or both) as well.

We know that it was interrupted at various times and there are no sources for some periods, so we don’t know its status. But there are records of it even after the fall of the Western Roman Empire. Theodoric arranged 120,000 modii of grain annually for the city of Rome, and Justinian ordered it to be continued in 554 AD. Rome remained an important symbolic city, and its ancient privileges such as the dole were upheld by successor kingdoms.

Around the reign of Pope Gregory I the old imperial official in charge of the dole disappears from records and the dole is soon after stopped in Constantinople. Both of these happened 600ish AD. In Rome, the Church stepped into the gap, first with the diaconia charitable organisations which were already providing grain in Ravenna and Naples. Later the papacy took direct control over the Roman dole. This was not entirely unexpected, because they had done the same thing during times of crisis in the past when the imperial or imperial legacy system broke down.

Sources: Jeffrey Richards: The Popes and Papacy in the Early Middle Ages; Jonathan Arnold: Theodoric, the Goths and the Restoration of the Roman Empire