In Catholic doctrine, is limbo eternal? The Catholic Church professes a belief in the existence of limbo. Although the doctrine on this subject has not yet been defined, does the Catholic Church believe that it is eternal?
We must distinguish between the two limbos:


Limbo of hell or of the Patriarchs (limbus inferni seu patrum)This is also known as Abraham's bosom. This limbo no longer exists (cf. "Reply to Objection 3" of this) because Christ has already descended into hell and brought those souls detained there to heaven during His Ascension.
Limbo for children (limbus puerorum)This is different than the limbo of the Fathers, which "is placed higher than the limbo of children." (cf. "I answer that…" of this). It is where the souls of unbaptized children are. It is a place of perpetual, natural happiness where souls are not actively punished by fire because they have not committed any personal sins; however, it is still hell, which, by definition, is the place where one is deprived of the Beatific Vision.


As St. Thomas Aquinas writes ("I answer that…" of this):


  The limbo of the Fathers and the limbo of children, without any doubt, differ as to the quality of punishment or reward. For children have no hope of the blessed life, as the Fathers in limbo had, in whom, moreover, shone forth the light of faith and grace. But as regards their situation, there is reason to believe that the place of both is the same; except that the limbo of the Fathers is placed higher than the limbo of children, just as we have stated in reference to limbo and hell (Article [5]).


Now, hell is eternal. The limbus puerorum is in hell. Therefore, the limbus puerorum is eternal.

Pope Pius VI condemned "as a Pelagian fable" the fact that the false Synod of Pistoia "introduced that middle place and state free of guilt and of punishment between the kingdom of God and eternal damnation.

The full quote of Pope Pius VI's “Auctorem fidei,” Aug. 28, 1794, is reproduced in Denzinger-Schönmetzer, Enchiridion Symbolorum, definitionum et declarationum de rebus fidei et morum. Ed. 34. 1965, §2626


  
  The doctrine which rejects as a Pelagian fable, that place of the lower regions (which the faithful generally designate by the name of the limbo of children) in which the souls of those departing with the sole guilt of original sin are punished with the punishment of the condemned, exclusive of the punishment of fire, just as it, by this very fact, that these who remove the punishment of fire introduced that middle place and state free of guilt and of punishment between the kingdom of God and eternal damnation, such as that about which the Pelagians idly talk,—false, rash, injurious to Catholic schools.
  


Where do these souls end up, then? They can't be in heaven or purgatory because they are not baptized, and they can't be punished with the fires of hell because they committed no personal sin; nor can they be somewhere between heaven and hell. Thus, they must be in hell (because they are deprived of the Beatific Vision, and hell is the place of being deprived of seeing God), but they are not punished by the fires of hell. This is the limbus puerorum.