The 23-year-old was part of a crew operating an armoured vehicle in a training area near the village of Enford on Salisbury Plain when the incident happened.
Sources say the vehicle overturned and hit a tree. Because there was live ammunition on board, there was a delay in rescuing the rest of the crew.
Emergency services had to bring in specialist equipment to right the vehicle and the army had to make it safe before the survivors could be extracted.
Wiltshire Police, the Army and the Health and Safety Executive are investigating the circumstances surrounding the soldier’s death.
A Wiltshire Police spokesman said: "Our thoughts are with the soldier's family and friends at this difficult time and we would ask that their privacy is respected."
A Dorset and Wiltshire Fire and Rescue spokesperson said: "We were called to Enford at 11.57am yesterday morning for a road traffic collision, and crews attended from Pewsey and Ludgershall, together with the heavy rescue unit from Trowbridge."
The MoD said: "It is with sadness that we can confirm the death of a soldier on Salisbury Plain Training Area.
"Wiltshire Police are investigating and therefore it would be inappropriate to comment further. The cause of death will be determined by the coroner.”
Devizes MP Danny Kruger said: "I was very sorry to hear of the death of a soldier on Salisbury Plain yesterday, during training near Enford.
"My sincere condolences to the soldier's family and to his comrades, and my thanks to the police and fire service for their prompt attendance at the scene.
"While thankfully rare, it is vital that all serious accidents that take place during military training exercises are comprehensively investigated.
"We owe so much to the young men and women who risk their lives for our safety and we must do everything we can to keep them safe as well."