Yes, I know “Dublin” is the obvious answer 
But I’m interested to know what the proper Irish pronunciation is. I seem to remember being told it’s something like “Blah Clear”, but isn’t “Baile” pronounced roughly as “Bally”, hence the profusion of towns called Ballywhatsit (in the English spelling)?
Obviously there is some confusion here. Googling Cliath pronounced gives at least half a dozen different versions:
and so on and so on… 
Well, there can be different pronunciations depending on the speaker’s accent, dialect, etc, although some of the ones you listed are a bit … bizarre, to my (virtual) ears. “Blah-klee-ah” would be the way I’d pronounce it.
“bala”, more like. Frequently the English spellings bear only slight relation to the Irish pronunciations.
Yes, many Irish words not in Gaelic become a bit funny when translated into English. Ballyfermot is apparently ‘train’ in English (lol) and ‘Baile Formaid’ or ‘Baile Diarmuid’ in Irish Gaelic.
My surname, English ‘Mickleburgh’ is ‘Mac Giolla Brighde’ (son of Saint Bridget) in Irish Gaelic, ‘Mac Giolle’ (son of the Saint) being the surname ‘Fitzpatrick’ in English; this because Irish for ‘son’ is ‘Fritz’ and the Patron Saint of Ireland is Patrick. Based on this, you’d think that ‘Mac Giolla Brighde’ would translate to be ‘Fitzbridget’ which sounds pretty stupid, but it went through a bit of a Yorke (traditional old language of Yorkshire, especially the West Ridings, which still exists as it wasn’t affected when all the Europeans came over in the Dark Ages and created English as we know it today. Sadly, Yorke, like Irish Gaelic, is becoming used less and less with people just speaking English.) make-over. Or something like that. The meaning (with bits of Irish, Irish Gaelic, Anglo, Yorke and possibly some Scottish, too) is something like Brave ruler over the tall mountains and leader of West Yorkshire. Or there abouts.
While keeping in mind that it will be pronounced differently by different speakers, and that any attempt to indicate Irish pronunciation by reference to English sounds is only an approximation, there are 2 options:
Pronouncing the words fully as
Boll-yeh aw-ha clee-ah
Running the words together as
Blaw clee-ah
The latter is how the name of the city is normally pronounced.
I’m afraid you have been VERY extensively misinformed.
+1 except my version of running the words together comes out more like
bol-yaw clee-ah 
And how.
A follow up question. Why is Baile Áth Cliath used as the official Irish version of Dublin and not Dubh Linn, which is equally Irish? Is it because the origin of that name is Norse?
Yikes! Haunted by a zombie of my own creation. (And an Irish one at that.)
I’ve never heard it pronounced it anything like the second one. The first is a close approximation of how it should be pronounced.
That second one is how it’s pronounced in Connemara anyway. I vary between the two.
As far as I’ve been able to ascertain it’s because there were originally two settlements, one Baile Átha Cliath (literally town of the ford of the wattles) which was a Gaelic settlement and may have been pre-Viking, and Dubh Linn (literally Black Pool), which became the Viking stronghold in the region.
These settlements lay close to each other in what is now central Dublin. The English name for the city of course derived from Dubh Linn (via Dyflin in Norse) but Irish speakers continued to call the bigger town and city that developed Baile Átha Cliath after the original Gaelic settlement. In so far as Dubh Linn means anything now, it only refers to that historical settlement as far as I know. I must warn you I’m not 100% on it and it could well be that I’m mistaken in some of the details. To complicate matters a bit I’ve noticed on wiki it’s claimed Dublin is occasionally referred to as Duibhlinn in Irish, that’s the first I’ve heard of that.
What I haven’t been able to ascertain is if Baile Átha Cliath has been in continuous use down the years as the Irish name for the city or whether it was reaffirmed when Irish placenames became common on signage post independence. And if this is the case, whether there was ever a latter day debate over the merits of Dubh Linn versus BAC. Perhaps Dr. Drake or another doper might know more.
That’s how it’s pronounced on the radio and TV news. 3 syllables.