Cyprian actually rejected the accusation that he believed in rebaptism because he considered only the baptism within the church to be a valid or true baptism.
Cyprian of Carthage is one of the most notable Latin theologians of the first five centuries alongside Tertullian and Augustine. For the first half of his life, Cyprian was a rhetorician, however, shortly after converting to Christianity under the guidance of Tertullian, he was elected bishop of Carthage in 249 CE. It wasn't long after being elected to the college of bishops that the catholic church across the Roman Empire endured the first, and one of the most severe of persecutions.
In June of 250, Emperor Decius' efforts to restore glory to Rome came to fruition in the form of mandatory cultic sacrifice to the Roman gods. All citizens were required to offer incense or a burnt sacrifice to the cultus of Rome. All who resisted were threatened with the loss of property and privilege. Clergy and bishops were intentionally targeted and tortured until fealty was sworn to Rome and the Roman pantheon. Sadly, many succumbed to this persecution and became apostate.
The large number of laymen and clergy who abandoned their faith and fellow Christians created a crisis for the church, as church leaders wondered how specifically to define "one, holy, catholic, and apostolic church" as we now confess according to the Nicene Creed. Practically speaking, early church bishops began to debate what to do with these apostate believers.
Two schools of thought arose: one was considered lax and the other rigorous. Those who promoted the more relaxed approach proposed simple private confession, absolution, and the laying on of hands by confessors to receive back those who had become apostate. The rigorist side denied that simple absolution and laying on of hands was enough. Instead, they argued the only way one could return to the church was by doing extreme acts of penance until death in hopes this would be enough to receive God's forgiveness. Only this, or being martyred for the faith during subsequent persecution, afforded the opportunity for salvation. A final and widely accepted opinion among the more rigorous bishops was that both those who lapsed as well as any Christian who had received baptism from the hands of bishops or clergy who had lapsed needed to be "re"-baptized to enter the church.
Cyprian was one such bishop who was accused of promoting the "re"-baptism of the lapsed. The reason for this lies in how Cyprian defines the church. For Cyprian, the church is a visible, external, association of the saints, under the episcopal leadership of the successors of Peter. The office of bishop is central. The bishop is in the church, and the church is in the bishop. The unity of the church exists in the succession of bishops from the first apostles. Apostolic succession demonstrated the genealogy of the church, so to speak. Likewise, the purity of the church exists in the character and holiness of the bishops.
Those bishops, clergy, and laypeople who became apostate, therefore, placed themselves outside the church. By abandoning God, they had aligned themselves with the devil. Instead of being instruments of Christ, they had become tools of the anti-Christ. Since baptism is the church's unique property, Cyprian suggested that lapsed bishops had illegal possession of the Sacrament. Therefore, baptisms that happened outside the church were, in fact, no baptisms at all.
For this reason, Cyprian actually rejected the accusation that he believed in rebaptism because he considered only the baptism within the church to be a valid or true baptism. He claimed, "He can no longer have God for his Father who has not the church for his mother," and even more famously, "There is no salvation outside the church." For those lapsed to receive the Holy Spirit and be forgiven, they needed to receive the true washing of regeneration and gift of the Spirit in the waters of baptism.
Cyprian's greatest opponent representing the lax view was Stephen the Bishop of Rome. However, the controversy never truly came to an end between the two opponents. Stephen of Rome threatened to excommunicate Cyprian over the disagreement but died before he could make good on his threat, and a year later, Cyprian would die a martyr's death.
The controversy would resurface with the emergence of Donatus, who, citing Cyprian, claimed that Sacraments performed by heretical/lapsed priests were invalid. Against Donatus and the Donatists., St. Augustine of Hippo claimed that it was not the character of the priesthood that sanctified the Church but the Holy Spirit who makes the church holy through the word and the Sacrament. It is the promise of God, not the work of man, that creates and sustains Christians and on the lips of sinners that the message of salvation comes. The view of St. Augustine over against Donatus is, to this day, confessed by the church catholic. One place this is made clear is in Article VIII of the Augsburg Confession, where the Donatists are not only condemned, but the church is uniquely defined according to the gospel.