Lyn - Ist century AD isn't really 'Dark Ages'....and as a term, Dark Ages is falling out of use in favour of more meaningful descriptions such as Migrations Period, Early Middle Ages etc.
In the time of the pagan Roman Empire - say up to about 400 as a broad sweep - libraries were kept as annexes of temples, and were also often owned by rich men, either because they were philosophers or to display.
The christian takeover was gradual and occasionally involved destroying non-Christian texts, but this varied hugely over time and place.
Byzantine rulers and monasteries continued the practice of keeping libraries but were pretty harsh on the whole on 'pagan' texts - this led to many learned men leaving Byzantium for Persia, taking their collections of writings with them.
In the west of Europe, christian monasteries were copying texts from the earliest times, even at very remote sites like Skellig Michael in the extreme west of Ireland.
During the early middle ages many of the incoming groups such as Goths, Saxons and Franks were characterised by their desire to be like the Romans. This included writing 'like the Romans' ie in books on parchment, and educating rich boys to read versions of Latin texts, so the copying of antique texts by monks continued.
Local events also affect the survival of texts like Ovid - in England for example the Dissolution of the Monasteries under Henry VIII caused unimaginable losses of ancient writings as did the Revolution in France.