grammaticality - Using articles before titles beginning with articles

What are the rules for the use of articles before titles beginning with articles? Which of the following sentences is correct?

1. I went to the The New York Times office this morning.


2. I went to The New York Times office this morning.


3. I went to the New York Times office this morning.


4. I am reading a The New York Times article.


5. I am reading a New York Times article.

Are "the The" and "a The" grammatical? If not, which is the article that should be dropped? Or is it simply a question of style?

According to Google, there are 90.3 million instances of "the The New York Times" on the Internet including 340,000 hits within the nytimes.com domain. I'm sure that there are plenty of false positives, but there are genuine cases too such as in the following sentence:

Writers from the The New York Times Dining section share Thanksgiving memories.