Stone was at the game with his young son, who watched as his dad tumbled over the outfield railing after catching the ball.
Arlington Fire Department officials said Stone was conscious after the fall. He was pronounced dead at a Fort Worth hospital less than an hour after he fell.
"We had a very tragic accident tonight and one of our fans lost their life reaching over the rail trying to get a ball," team president Nolan Ryan said somberly after the Rangers' 6-0 victory over Oakland. "As an organization, and as our team members and our staff, we're very heavy-hearted about this, and our thoughts and prayers go out to the family."
Ronnie Hargis was sitting next to Stone in the first row of seats in left field. The men had been talking before the accident. Hargis reached out to try to grab Stone, who fell headfirst through a gap of several feet that is between the seats and the 14-foot(4-meter)-high outfield wall.
"He went straight down. I tried to grab him, but I couldn't," Hargis said. "I tried to slow him down a little bit."
The area where Stone landed was out of sight from the field.
It is the second fatal fall at a Major League Baseball stadium this season. In May, a 27-year-old man died after he fell about 20 feet (6 meters) and struck his head on concrete during a Colorado Rockies game. Witnesses told police that the man had been trying to slide down a staircase railing at Coors Field and lost his balance during a game.
There was an audible gasp in the stands at Rangers Ballpark when Stone tumbled over the rail, eerily similar to an accident there almost exactly a year earlier. Another firefighter fell about 30 feet (9 meters) from the second-deck of seats down the right-field line while trying to catch a foul ball on July 6, 2010. Tyler Morris, a firefighter from the Lake Cities Fire Department near Dallas, sustained severe injuries last year when he dropped onto seats where other fans were sitting.
The latest accident happened in the second inning after Oakland's Conor Jackson hit a foul ball that ricocheted into left field. Hamilton, the reigning American League MVP elected by fans to start his fourth consecutive All-Star game next week, retrieved the ball and tossed it into the stands as players routinely do.
Safawna Dunn, who was sitting nearby, said Stone was calling for Hamilton to throw him the ball.
Ryan described Hamilton as being "very distraught over this, as the entire team is."
Former President George W. Bush, who used to be the team's managing general partner and often attends games, was sitting in the front row with Ryan near the Texas dugout when the accident happened. Ryan left moments later while Bush remained in the seats.
Ryan said Bush was aware of what was happening.