Ugh. No. I’m wrong.
I’m going with August 17. The it can’t be the 18th or 19th, because then Bernard would have known the answer because they’re unique. Since Albert knows Bernard doesn’t know, he must know that the month isn’t May or June. So all those are out. Once he says he still doesn’t know, but Bernard know does know, it must be a unique date from July or August, and August 17th is the only possibility.