CathInfo.com

I believe they go to limbo.

Same here. From Who Shall Ascend?:

Likewise, there is a minimum damnation and a maximum, as damnation means, essentially, the lost of the Beatific Vision. Minimum damnation is death in infancy without Baptism and eternity in Limbo. Deprivation of the vision of God is damnation; in the case of infants, it is the punishment which they share with all men for the sin of Adam. Thus Pope Innocent III (1198-1216) said: "The penalty of original sin is the loss of the vision of God; the penalty of actual sin is the torment of everlasting hell."

And St. Bonaventure said: "The punishment of being deprived of the sight of God and the loss of heavenly glory effects both adults and children who are unbaptized. The children are punished with others, but by the mildest punishment because they deserve only the punishment of loss but not the punishment of the senses." Any attendant sufferings are a matter of the degree and are determined by the strength of the sinner's obdurate rejection of God.

Unbaptized babies who die are still guilty of original sin, and on that account, they cannot be saved. It is commonly believed that they go to a place of Hell, named Limbo, where there is not physical suffering but "natural" happiness (still, this is a place where God is not, and they do not enjoy the Beatific Vision). There are some, like st. Augustine. who believed unbaptized babies who die are not except from actually suffering torments in Hel, just like everyone else, including physical pain. What we do know for sure is that the Church infallibly defined at the ecuмenical councils of Lyons and Florence, that the guilt of original sin suffices for damnation.

“The souls of those who die in mortal sin or with original sin only, however, immediately descend to hell, to be punished moreover with disparate punishments. […] They will go into the everlasting fire prepared for the devil and his angels.” (Florence)

Liberals and other sentimentalists such as cushinguites of course want the unbaptized babies who die to enter Heaven. "An innocent baby cannot go to Hell, that would not be an act of merciful God!", they say. They forget that all of us are actually born into a state of natural damnation (because of the original sin) and not one of us is innocent. All of us are guilty. These liberals pretend that these babies can be also saved through the "baptism of desire"; or that the "desire" of the mother for him to be saved would be sufficient.

Completely absurd.