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.

So maybe Limbo is like the 'garden of eden'?  An earthly paradise, but far below heavenly paradise.

Doesn't the Church teach that unbaptised souls go to Limbo, a place of perfect natural happiness.

I recall Pope John Paul II bypassed this teaching and suggested they all go to heaven.

I was at a pro-life convention when I brought up the matter of Catholic teaching as a means to deter Christians from even thinking about abortion.
I said that if Pope John Paul II's 'wish' were true, what were Catholics worried about when it came to abortion. I said every abortion sent a child to heaven, a child that if it lived may have ended up in hell.

Well, as you can guess there was silence in that room of a few hundred people, including priests. The chairman said that a Fr Whatshisname would see me after the meeting and correct my thoughts.

Needless to say Fr Whatshisname never came within a mile of me afterwards. I have never attended a pro-life meeting after that.

They don't go to heaven is the only answer to give.

I agree they don't go to heaven, but I asked the question to see what others think. Many people who call themselves Catholics believe they go to heaven or at least that some of them do. Even some who claim to be trads believe this and after being reminded of this I started this thread to see what the Cathinfo posters believed.

It is not a pelagian fable that those who depart this life with original sin only (i.e. unbaptized infants) are condemned, but without the punishment of fire. Pelagians were also wrong that there is a place between Heaven and hell free of guilt and punishment.

This is an infallible quote by Pope Zosimus (confirming Canon 2 of Council of Carthage):

It has been decided that, likewise, if anyone say that it might be understood that in the kingdom of Heaven, there will be some middle place or some place anywhere infants live who departed from this life without Baptism, without which they cannot enter the kingdom of Heaven, which is life eternal, let him be anathema. For the Lord says:"unless a man is born again of water and of the Holy Ghost, he shall not enter the kingdom of God"

This is the same Pope, St. Zosimus, who also said "No one of our children is held not guilty until he is freed through Baptism".

Also from this infallible pronouncement:

" The punishment of original sin is the loss of the vision of God; the punishment for actual sin is the torments of everlasting Hell".

We can deduce that those who die in the state of original sin like unbaptized infants do NOT see God, as they incur the "loss of the vision of God"; but because these babes do not have any actual sin, then they do not have any "torments of everlasting Hell".