Um no. There is no official doctrine on that matter. There are quite a collection of semi official ones. The two most popular however are limbo (which may or may not be escapable but should contain and interesting selection of people) and/or prayerful hope that god sorts it out somehow.
Actually, you do realize that Limbo is the outer circle of Hell, right? It's not Purgatory or anything. And at least for Catholics the dogma has always been that unbaptized babies will undergo _some_ form of punishment for that, just, you know, not the worst.
For a more in detail description of it and the theological history of it, try the catholic encyclopaedia:
http://www.newadvent.org/cathen/09256a.htm
You'll notice that for most of history, starting with St Augustine, essentially the dogma was that they'll fry. And at least one pope actually publicly denounced the heresy that fire wouldn't be involved for babies. (Though he did not invoke infallibility on that one apparently.) But even that heresy still involved _some_ punishment, just, you know, they didn't want babies to fry.
And while in the meantime milder interpretations have become more popular, technically nobody rejected the old ones either. Essentially the church fell back to "ugh, we don't really know, after all."
Finally, you'll notice that Limbo is _permanent_ for unbaptized people. There's no prayer that will get those babies outta there.
But really, either way, we're talking a god who'd look at that poor, severely premature born girl -- if she had died in the process -- and essentially decide that she's damned (to a debatable extent) by original sin and lack of baptism. And totally not worthy of heaven.
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