First, "limbo" can refer to two distinct places, so we have to be clear from the get go exactly what we are talking about. "Limbo" can be either:
1) "The Limbo of the Fathers" - the place where the righteous dead went before Christ died and opened the gates of heaven for them. This is in the Bible - see 1 Peter 3:19. This Limbo surely existed, although there would be no one in it for the last two millennia.
2) "The Limbo of Infants" - the place where unbaptized babies go. This isn't explicitly mentioned in Scripture (but neither is the Trinity, so that isn't an argument against it). It can, however, be inferred from the doctrine that all must be baptized to go to heaven (see John 3:3).
Your question concerns "the limbo of infants."
It might be interesting to look at the word "limbo" and see what that tells us. The English word "limbo" comes from the Latin "limbus", meaning "border" or even "hem". Thus, limbo would refer to a place at the border of, but not inside of Hell. The souls in limbo, therefore, are not numbered amongst the damned.
Although not popular today, some of the greatest minds in Church History have taught that unbaptized babies go to limbo, including Dante Alighieri, the greatest poet and lay scholar in Church history; St. Thomas Aquinas, the greatest doctor in the history of the Church (see HERE); and various Church Fathers including St. Gregory Nazianzen , Tertullian, and St. Ambrose. Other great Catholics, including the second greatest doctor in the history of the Church, St. Augustine didn't teach limbo (Augustine taught that unbaptized infants would go to Hell, but have the mildest of all punishments).
More recently, especially in the twentieth century (even before Vatican Council 2), theologians began contemplating the idea that the unbaptized go to Heaven. The Church has never officially endorsed this position. The Congregation for the Doctrine of the Faith (then called simply the Holy Office) issued a statement in 1958 urging parents to baptize their children ASAP, noting that the state of the unbaptized after death is unknown,
"The practice has arisen in some places of delaying the conferring of Baptism for so-called reasons of convenience or of a liturgical nature" a practice favored by some opinions, lacking solid foundation, concerning the eternal salvation of infants who die without Baptism. Therefore this Supreme Congregation, with the approval of the Holy Father, warns the faithful that infants are to be baptized as soon as possible..."The Catechism of the Catholic Church likewise leaves the door open stating, in paragraph 1261:
"As regards children who have died without Baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them. Indeed, the great mercy of God who desires that all men should be saved, and Jesus" tenderness toward children which caused him to say, "Let the children come to me, do not hinder them," allow us to hope that there is a way of salvation for children who have died without Baptism. All the more urgent is the Church's call not to prevent little children coming to Christ through the gift of holy Baptism"
The closest the Church has ever gotten to making a definitive statement on limbo was with Pope Pius VI's Auctorem Fidei in which he condemned the Jansenists for teaching, as certain, Augustine's theory that the unbaptized were in Hell, but were not tormented by the flames of Hell. Pius doesn't declare that limbo exists, he simply rebukes the Jansenists for teaching that limbo doesn't exist, Pius, like the Church today, was leaving the question open.
Unfortunately, many Catholics today think that Limbo has been definitively rejected in favor of the idea that all unbaptized babies are certainly saved (which dovetails with an increasing tendency toward universalism, that all will be saved, in the minds of many Catholics). This simply isn't the teaching of the Church. The fact is, we really don't know what will happen to them and Catholics are free to decide for themselves between Augustine's theory (they go to Hell), Aquinas' (they go to limbo), and simply trusting in God's mercy (they go to Heaven).