In case someone asks me later and I forgot(!), this is the solution:
Dialogue can be reduced to:
Ben says: "I know that Art doesn't know"
Art says: "Now I know"
Ben says: "Ah, now I also know"
(notice that there is only one "2" and one "7")
Ben says: "I know that Art doesn't know"
Eliminates Jun and Dec, else Ben wouldn't be sure,
since Art could have Jun 7 or Dec 2.
Art says: "Now I know"
Knowing now that month is Jan or Sep, Jan 5 and Sep 5 are eliminated,
since being in both months, Art would not know if he was given 5.
Ben says: "Ah, now I also know"
Only way Ben can know is if he was given Sep, since only "1" is possible;
if given Jan, he'd have the 4th and 8th as possibilities.
So it's Sep 1.