WHERE: Overheard in the next office.
CORRECTION: This is NAME WITHHELD, calling from the Korean Mission.
GRAMMAR: Ah, articles! Is there any part of speech more vexing for non-native speakers whose native languages don't use them? Prepositions, maybe — try explaining why we get in an airplane but on a bus — but prepositions at least have the decency to be vexing to native speakers as well. (Are you waiting for a friend who's on line, or on a friend who's in line?) With articles, any native speaker knows what goes where, but just try explaining why.
A good rule is that you need the if both parties know which one you mean — the post office, the beach — or if there is only one around — the sun, the United Nations.
In this case, both of these factors are true of the noun Korean Mission, but it's particularly the latter factor that counts: there's only one South Korean Mission to the UN, and both parties know the phrase Korean Mission is an abbreviation of that longer term. As such, a definite article is needed.
BONUS: Why is it a definite article is needed rather than the definite article is needed in the preceding sentence? You could probably go either way, but which you choose depends on how many definite articles you think there are. In this case, I'm counting each instance of the as another definite article, so a specifies that definite article is one of many. But you could just as easily count definite article as a singular categorical noun, in which case it would take the because there's only one around. (We're not like those profligate Spaniards, with their varieties of singular and plural articles in various genders!)