Hawkman joins the JLA!
The story: "Riddle of the Runaway Room!" by Gardner Fox, Mike Sekowsky, and Bernard Sachs. Two-bit crook Joe Parry finds a device that came from outer space which pretty much creates anything one asks of it. He tries to commit crimes with it which gets the JLA involved. Luckily new member Hawkman saves the day!
Roll Call: Superman, Batman, Wonder Woman, Aquaman, Flash, Green Lantern, Martian Manhunter, Green Arrow, Atom, and new member Hawkman!
Notable Moments: The JLA has already decided to induct Hawkman by the time the issue opens, so we are spared Flash's tiresome proposal of Adam Strange as the new member.
There's a charming-in-its-out-of-date-ness scene where the Atom has to carefully explain to Hawkgirl why Hawkman is being invited to join, but not her. She cheerfully says its ok because, since Hawkman is the leader of their team, it counts as a honor for both of them. Now go home and bake some cookies, Shayera! You know what they say--tie a woman to the bed with just enough room to get to the Absorbascon.
I would argue that with this issue, the JLA really was the cream of the crop of the DCU--each member had his own book(s) or strip, and each of them were iconic in their own way. The membership at this point would stay consistent for the next forty issues or so.
Nothing against future members Black Canary, Elongated Man, Red Tornado, Hawkgirl, Zatanna, and Firestorm, but I think by the time the JLA got to them, the JLA itself was legendary in the DCU, and by adding a new member, the team bestowed prestige upon the newly-chosen hero, not the other way around.