Not quite. First of all, the rule evaluates from inside out, deepest genitive first, so right-to-left. A definite to the right obviates the need for one to the left, but not the other way round.

(In fact, it was suggested to me last week that this rule is taught completely wrong in Scottish Gaelic, and I suspect the same holds for Irish -- the rule I was told last week is that only the leftmost, genitive marked noun can take the article, and everything else is qualified by the noun below it (analogous to how you can't say "John's brother's the son is John's the nephew in English", although word order here is reversed).)

Now... is it possible to have two rightmost nouns? Actually yes, and it's best shown by breaking the phrase down into its components.

Consider that the following is not a subpart of the nounphrase:
.. Dé na Laethanta Deireanacha
It is not referring to a "God of the final days", so "God" is leftmost in a subphrase, and "the final days" is leftmost in the full phrase.

Eaglais Shoiscéalaíochta na Laochra Paidreoireachta Idirnáisiúnta: Arm Neartmhar Ungtha Dé dena Laethanta Deireanacha

-Eaglais is feminine

-Laochra: As far as I know, broad endings don't tend to lenite the following word.

-Dena: I think having dena makes the sense clearer?

That was a mouthful, in the end it was hard to know who owned what .

Just from looking at your emblem http://www.iglesiainternacional.cl/engl ... ?photoId=8 , it seems as if the "Evangelical Church" part is a separate entity (a seperate sentence) to the "international prayer warriors".

If you are trying to recreate this emblem in Irish it would go as follows, reading from top to bottom:

-Laochra Paidreoireachta Idirnáisiúnta-

-Greasán Domhanda- 'Worldwide Network'

-Ie é Íosa an Tiarna- 'Jesus is Lord'

-An Airm Ollchumhachtach/ Neartmhar Ungtha Dé dena Laethanata Deireanacha-

-Eaglais Shoiscéalaíochta-

Like NiallBeag, I think 'de' can't be used here, and that the two genitives are okay (though I got a bit lost with the "rightmost/leftmost" directions). As he says, it isn't "the army of the endtime God" - which would be arm Dhia na laethanta deireanacha; it's "the endtime army of God", which would be arm De na laethanta deireanacha. That's my understanding of the rule anyway, but I'm certainly open to correction.

So, Chevy, I suggest:

Arm ungtha mórchumhachtach Dé na laethanta deireanacha (which, apart from 'mórchumhachtach', is what Gumbi had). But get it confirmed first.

'Mighty army' is translated as 'arm mórchumhachtach, in 'An Foclóir Nua', but 'neartmhar' would do too.