The agency confirmed the shooting at the Ysleta Border Patrol Station at about 12:45 p.m. Tuesday.
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No other injuries were reported.
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Two Border Patrol officials told ABC News the firearm discharged belonged to an agent, but it's unclear what led to the shots being fired.
The person in custody was taken to the hospital, where that person died.
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It's not immediately known why the person was in custody.
The FBI, El Paso police and CBP's Office of Professional Responsibility are investigating the incident, with the bureau leading.