Actually, the requirement for a Sacramental marriage is that both parties be baptized, and marry according to the Church's *definition* of marriage, not the Church's Rite of Marriage. Just as a priest is not necessary to perform a Baptism, a Baptized couple confers the sacrament of marriage on *each other*. The priest does not confect the sacrament and is merely a witness for the Church. A 'natural contract' occurs when one or both of the parties are unBaptized...not non-Catholic.
All that being said...yes, the marriage still needs to be recognized by the Church, if the couple intends to be participate in the life of the Church, and children are required to be raised in the faith.
I recall that I had to sign a paper regarding that when my husband converted.