Profane actually comes from the Latin profanus (literally "out of the temple"), which served as the opposite of sacer, "sacred." Like the English profane, profanus was used to mean both secular and impious, depending on the context.

A good example of the Roman treatment of the sacred/profane dichotomy is the concept of dies fasti/nefasti. Dies fasti were days on which it was permitted to conduct public business, including governmental and judicial business, while dies nefasti were festival days on which business was not permitted. The terms are related to fas, which means "that which is permitted by divine law", and nefas, which is its opposite.

Sources: Oxford Latin Dictionary, Oxford Classical Dictionary