Can the Church Perform a "Secret Marriage"?

Q. Can the Catholic Church perform a purely sacramental marriage without civil effect?

A. A “secret marriage” without civil effects, or an “extraordinary form” celebration of marriage without civil effects, could be allowed by the local bishop on a case-by-case basis. But currently in the U.S. the bishops have deemed it imprudent to do so. It would be illegal according to various state laws throughout the U.S., and for that reason the bishops hesitate. (In some states it could be a Class 4 Felony to marry a couple without a marriage license.)

The priest is not allowed to perform/witness such a marriage without the approval of the bishop. (The relevant canons are Canons 22, 1672 and 1059.)