Stone fell around 7:30 p.m. during the second inning of the Rangers-Oakland A's game, KDAF-TV, Dallas/Fort Worth, reported.
Witnesses and the Arlington Fire Department said Stone, 39, lunged to catch a foul ball outfielder Josh Hamilton threw into the center field stands.
He fell 20 feet head first over the rail onto concrete. Another fan saw Stone leaning over the rail and grabbed his shirt but could not hold on to him.
Stone was taken to John Peter Smith Hospital in Fort Worth but went into cardiac arrest on the way and was pronounced dead around 8:30 p.m.
Rangers President Nolan Ryan said: "We are deeply saddened to learn that the man who fell has passed away as a result of this tragic accident. Our thoughts and prayers are with his family."
Ryan said Hamilton was obviously upset over the incident.
"I think as any of us would be," Ryan said. "Josh is very distraught over this, as is the entire team."