[rd_aaron]

I think I got it:

Spoiler!

Answer: July 16

It can't be May or June, because Bernard doesn't know at first, and he'd know the birthdate if it was May 19 or June 18 (only one "18" and one "19).

Bernard couldn't have been given 14 because he wouldn't know if it was July or August. He knows the birthdate because the other number (either "15", "16", or "17") was eliminated in May & June. So that gives us July 16, August 15, or August 17.

For Albert to now know the answer, it would have to be July 16 since July 14th was eliminated as per the above. It can't be August 15th or 17th because Albert wouldn't have been able to figure out which of those two it was.