See already sigma 336.
[1] Procopius, History of the Wars of Justinian 1.20.9 (here somewhat rearranged; cf. web address 1). Later on, Procopius tells more about Roman efforts to obtain silk (History of the Wars of Justinian 8.17 (web address 2)): "About the same time [ca.550 CE] there came from India certain monks; and when they had satisfied Justinian Augustus that the Romans no longer should buy silk from the Persians, they promised the emperor in an interview that they would provide the materials for making silk so that never should the Romans seek business of this kind from their enemy the Persians, or from any other people whatsoever. They said that they were formerly in Serinda, which they call the region frequented by the people of the Indies, and there they learned perfectly the art of making silk. Moreover, to the emperor who plied them with many questions as to whether he might have the secret, the monks replied that certain worms were manufacturers of silk, nature itself forcing them to keep always at work; the worms could certainly not be brought here alive, but they could be grown easily and without difficulty; the eggs of single hatchings are innumerable; as soon as they are laid men cover them with dung and keep them warm for as long as it is necessary so that they produce insects. When they had announced these tidings, led on by liberal promises of the emperor to prove the fact, they returned to India. When they had brought the eggs to Byzantium, the method having been learned, as I have said, they changed them by metamorphosis into worms which feed on the leaves of mulberry. Thus began the art of making silk from that time on in the Roman Empire."
[2] e.g. in 39.9 of the Periplous of the Red Sea attributed to Arrian.
[3] cf. sigma 336.