Àn bhfuil sin ceart? History of Dublin - Stair na Baile Átha Cliath

With place names, you just lenite the first letter to form the genitive, so I think it should be:

Stair Bhaile Átha Cliath

But be sure to wait for confirmation of that ...

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

Breandán's translation is certainly how I would write it.

A Bhreandán:

When do you use the "na"? I've never really figured that out.

Féabar

You use "na" when the word is feminine singular (or plural, any gender) when it's preceded by an article.
Dublin is "Baile Átha Cliath", not "an Bhaile Átha Cliath" (btw it's masculine).

You'd say:

stair na Gaeilge because it's an Ghaeilge

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Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

As Lughaidh has already said, na is used with the genitive singular of feminine nouns and the genitive plural of nouns of either gender.

The only time you would normally use na with a place name is if the name of the place starts with an article and is feminine singular or is plural.

Countries are usually feminine and often start with an article, e.g., an Fhrainc, an Spáinn, etc., so you often see na in the genitive of a country name, i.e., na Fraince, na Spáinne, etc.

One exception to that general rule is Éire, which doesn't have an article in the nominative but becomes na hÉireann in the genitive, i.e., ceol na hÉireann, mná na hÉireann, etc.

Here's a list of other country names:

http://en.wikipedia.org/wiki/List_of_Irish_exonyms

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).

(I agree). And yet Saorstát Éireann was written on coins at one point.

Maybe they were pressed for space.

One thing that is certain about the Irish language: if you have a general "rule", there will be exceptions ... and then exceptions to those exceptions ... and probably a few exceptions to those exceptions ... Let it never be said that the Irish are not exceptional people.

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Múinteoir Gaeilge - Irish Teacher
My "specialty" is Connemara Irish, particularly Cois Fhairrge dialect, but I can also speak Ulster and Munster Irish with native-level pronunciation.
Is fearr Gaeilge ḃriste ná Béarla cliste, cinnte, aċ i ḃfad níos fearr aríst í Gaeilge ḃinn ḃeo na nGaeltaċtaí.
Gaeilge Chonnacht (GC), go háraid Gaeilge Chois Fhairrge (GCF), Gaeilic Uladh (GU), Gaelainn na Mumhan (GM), agus Gaeilge an Chaighdeáin Oifigiúil (CO).