Objection 1. It seems that one that is not baptized cannot confer the sacrament of Baptism. For "none gives what he has not." But a non-baptized person has not the sacrament of Baptism. Therefore he cannot give it.
Objection 2. Further, a man confers the sacrament of Baptism inasmuch as he is a minister of the Church. But one that is not baptized, belongs nowise to the Church, i.e. neither really nor sacramentally. Therefore he cannot confer the sacrament of Baptism.
Objection 3. Further, it is more to confer a sacrament than to receive it. But one that is not baptized, cannot receive the other sacraments. Much less, therefore, can he confer any sacrament.
On the contrary, Isidore says: "The Roman Pontiff does not consider it to be the man who baptizes, but that the Holy Ghost confers the grace of Baptism, though he that baptizes be a pagan." But he who is baptized, is not called a pagan. Therefore he who is not baptized can confer the sacrament of Baptism.
I answer that, Augustine left this question without deciding it. For he says (Contra Ep. Parmen. ii): "This is indeed another question, whether even those can baptize who were never Christians; nor should anything be rashly asserted hereupon, without the authority of a sacred council such as suffices for so great a matter." But afterwards it was decided by the Church that the unbaptized, whether Jews or pagans, can confer the sacrament of Baptism, provided they baptize in the form of the Church. Wherefore Pope Nicolas I replies to the questions propounded by the Bulgars: "You say that many in your country have been baptized by someone, whether Christian or pagan you know not. If these were baptized in the name of the Trinity, they must not be rebaptized." But if the form of the Church be not observed, the sacrament of Baptism is not conferred. And thus is to be explained what Gregory II [Gregory III] writes to Bishop Boniface: "Those whom you assert to have been baptized by pagans," namely, with a form not recognized by the Church, "we command you to rebaptize in the name of the Trinity." And the reason of this is that, just as on the part of the matter, as far as the essentials of the sacrament are concerned, any water will suffice, so, on the part of the minister, any man is competent. Consequently, an unbaptized person can baptize in a case of urgency. So that two unbaptized persons may baptize one another, one baptizing the other and being afterwards baptized by him: and each would receive not only the sacrament but also the reality of the sacrament. But if this were done outside a case of urgency, each would sin grievously, both the baptizer and the baptized, and thus the baptismal effect would be frustrated, although the sacrament itself would not be invalidated.
Reply to Objection 1. The man who baptizes offers but his outward ministration; whereas Christ it is Who baptizes inwardly, Who can use all men to whatever purpose He wills. Consequently, the unbaptized can baptize: because, as Pope Nicolas I says, "the Baptism is not theirs," i.e. the baptizers', "but His," i.e. Christ's.
Reply to Objection 2. He who is not baptized, though he belongs not to the Church either in reality or sacramentally, can nevertheless belong to her in intention and by similarity of action, namely, in so far as he intends to do what the Church does, and in baptizing observes the Church's form, and thus acts as the minister of Christ, Who did not confine His power to those that are baptized, as neither did He to the sacraments.
Reply to Objection 3. The other sacraments are not so necessary as Baptism. And therefore it is allowable that an unbaptized person should baptize rather than that he should receive other sacraments.
Source: Summa Theologica (New Advent)