CathInfo.com

After Agapetus's death on 22 April, 536, Vigilius return to Rome equipped with letters from the imperial Court and with money. Meanwhile Silverius had been made pope through the influence of the King of the Goths. Soon after this the Byzantine commander Belisarius garrisoned the city of Rome, which was, however, besieged again by the Goths. Vigilius gave Belisarius the letters from the Court of Constantinople, which recommended Vigilius himself for the Papal See. False accusations now led Belisarius to depose Silverius. Owing to the pressure exerted by the Byzantine commander, Vigilius was elected pope in place of Silverius and consecrated and enthroned on 29 March, 537. Vigilius brought it about that the unjustly deposed Silverius was put into his keeping where the late pope soon died from the harsh treatment he received. After the death of this predecessor Vigilius was recognized as pope by all the Roman clergy. Much in these accusations against Vigilius appears to be exaggerated, but the manner of his elevation to the See of Rome was not regular.

I take that to mean that Silverius was still the pope until he died.  Only after Silverius died did the Roman clergy recognize Vigilius as the pope.  I also understand that others will argue that Vigilius was the true pope before Silverius died but I don't think that could ever be certain.  What is certain is that the Roman clergy all accepted Vigilius as the true pope after Silverius died and that Vigilius was actually Catholic, unlike the pretender popes of the Conciliar sect.