Last month (Feb. 12, to be specific), Bishop John Kudrick of the Eparchy of Parma (Ohio) of the Ruthenian Byzantine Catholic Church in America ordained Fr. Joseph Marquis to the priesthood.
Fr. Joseph is, and has been for a number of years, a happily married man. He is now a Catholic priest, just as much a Catholic priest as any of the thousands of Catholic priests who serve in Roman Catholic parishes across the country (and the world!).
The Byzantine Catholic Church is in full communion with Rome (i.e., fully Catholic), meaning that any Catholic individual, Roman Catholic or otherwise, is absolutely free to partake of the Sacraments at the hands of Fr. Joseph. Attending a Mass (or, more properly, a Divine Liturgy, as it is referred to in the Catholic East) celebrated by Fr. Joseph would, in fact, satisfy your Sunday obligation as a Catholic. In short, Fr. Joseph is, in fact, a married Catholic priest.
Priestly celebacy is a discipline that is subscribed to by the Latin Church… it is not dogma throughout the entire Catholic Church.