Zeph, correct me if i'm wrong ... but wasn't Latin the dominant language during the Roman Empire(a part of Ancient Greece)? Which would be one of the main reasons why Greek and Latin (the two dominant languages of the RE) were "infused"?
"Ancient Greece" geographically was pretty much the same area as modern Greece today, though by the time of the Roman conquests in the 2nd and 1st centuries BCE, Greek was spoken in many places throughout the Mediterranean world, though mostly in the Eastern half.
I'm not sure what your notion of "Ancient Greece" is but saying the Roman Empire was a part of it, as opposed to the other way around, is as patently false.
Latin was the language of government and law under and throughout the ancient Roman Empire, and it spread as a spoken language to many parts of the Western half of the Empire, i.e. places where its descendants, the Romance languages, are spoken today.
In the Eastern half of the Roman Empire the most widely spoken language was, and remained to be, Greek all the way up to the period of the Muslim conquests in the seventh and eighth centuries CE.
This is not even taking into account the scores of other languages spoken in places under Roman rule, such as Punic in North Africa. The thing about empires is that just because you're ruled by a certain culture or nation, doesn't mean you automatically assimilate to that culture in all its aspects. The upper classes in all those provinces did to a higher degree, but that was a tiny minority of people.
Despite coming under Roman rule, the Greek language maintained its status as a language of elite culture, philosophy, science, literature, etc. and for a long time many native Roman writers wrote in Greek instead of Latin (the emperor Marcus Aurelius, for instance). It also continued as a living language of the common people as it had been for centuries before Roman rule, and continued to be throughout the Byzantine period.
Latin and Greek did not "infuse" nearly as much as you'd think, since Latin speakers in the West and Greek speakers in the East never truly blended together into a single culture. The elites became bilingual for the most part, though in the later period of the Roman Empire, when it became more and more politically divided between West and East, the two cultures diverged more and more and became estranged. Latin stopped being used as the language of government by the Eastern Roman Empire by the seventh century.