Ante-Nicene Fathers Volume 5
Appendix: A Treatise on Re-Baptism by an Anonymous Writer
Once again we're faced with a work by that most prolific of writers, "anonymous." And once again we're back to the question of legitimate baptism. This time the dispute is over whether or not those who were baptized into the "true" church, and then left for a schismatic church (Novatian's, for example), and then return, need to be re-baptized. If not, will a simple "laying on of hands" by the pastor and elders suffice?
The author in a long passage points out that this question has really been settled, but because of contentious human nature we refuse to acknowledge the authority of the church and so once again the Scripture arguments need to be brought up. Which is a great thing to notice: even though the decision has been made, force is not the recourse. Instead the authority of Scripture and persuasion are the tools in question, and remain so for centuries yet to come in the church.
By and large, this treatise can be skimmed. There's nothing here that wasn't also in the letters from Cyprian, other than some more explicit expositions of Scripture on baptism (which were largely only hinted at in Cyprian's letters). As there, we might not be able to go along with the exegesis or some of the theological conclusions--especially those that dance near baptismal regeneration and the idea of a single, unified church as the only legitimate one--but we can certainly go along with the submission to Scripture and the idea that we ought to be doing church and administering the sacraments correctly. Interestingly, here baptismal regeneration is explicitly rejected, "lest on this principle we should believe that even Gentiles and heretics, who abuse the name of Jesus, could attain unto salvation without the true and entire thing [that is, without faith]."
The overall point is that in baptism we have a great deal of flexibility, so long as we are being faithful and solemn in our administration of the sacrament. In answer to the specific question, no re-baptism is required, since the water does not save and is not necessary in the strictest sense...
So, this is a fine one to read but not because there's anything new to be picked up here.