Several historians told varying accounts of a Arab army led by Amr ibn al 'Aas sacking the city in 642 after the Byzantine army was defeated at the Battle of Heliopolis, and that the commander asked the caliph Umar what to do with the library. He gave the famous answer: "They will either contradict the Koran, in which case they are heresy, or they will agree with it, so they are superfluous."
Apparently the holding in the library at the time were still quite substantial. It should be noted that the (Christian) Byzantine Empire had controlled Alexandria for centuries prior to the conquest by the Arabs.
Ping to investigate later
It seems to be the consensus that the Caliph Omar incident is apocryphal.
As I recall, the scrolls from the library were so extensive, that they heated the public baths in Alexandria for 6 months as they burned.