Thanks for confirming that, Despina. I just changed my date from 5/31 to 6/15, and these guys posting asking if the June MCAT is too late were starting to stress me out.
Guys, if you want to have the least delay possible, have all of your transcripts in and submit your AMCAS application on the very first day, June 5th. It will probably take a few weeks to get verified anyway. And from what I've heard, a number of schools will send you secondaries before your MCAT score even comes out, so you can get started on some secondaries. Then, when your score comes out in mid-July and is transmitted to the schools, you'll be complete at some (those that you already returned the secondaries for), and perhaps just delayed by two weeks for the others if you turn around their secondaries asap. At least, that's my plan! Good luck.