Lance Corporal Craig Roberts died during a military exercise in the Brecon Beacons on Saturday when temperatures soared to 29.5C (85.1F).
A second soldier, who has not yet been named, also lost his life, while a third remains in hospital.
A total of six men were taken ill in the stifling conditions and an investigation into the circumstances has been launched.
In the statement, the MoD said: "It is with great sadness that the Ministry of Defence can confirm the death of L/Cpl Craig Roberts and another Army reserve soldier during a training exercise in the Brecon Beacons at the weekend.
"Our thoughts are with their families and friends at this difficult time.
"The families of the soldiers have requested that their privacy is respected at this difficult time."
It is not yet known whether the MoD will name the second soldier.
Police and the MoD are investigating the deaths of the men, who were part of a group training in Wales's rugged Brecon Beacons on Saturday.
Sources have said the men were at the start of a period of assessment in the selection process for the Territorial Army's section of the SAS, after completing a period of training and preparation.
Rest in Peace lads.
A second soldier, who has not yet been named, also lost his life, while a third remains in hospital.
A total of six men were taken ill in the stifling conditions and an investigation into the circumstances has been launched.
In the statement, the MoD said: "It is with great sadness that the Ministry of Defence can confirm the death of L/Cpl Craig Roberts and another Army reserve soldier during a training exercise in the Brecon Beacons at the weekend.
"Our thoughts are with their families and friends at this difficult time.
"The families of the soldiers have requested that their privacy is respected at this difficult time."
It is not yet known whether the MoD will name the second soldier.
Police and the MoD are investigating the deaths of the men, who were part of a group training in Wales's rugged Brecon Beacons on Saturday.
Sources have said the men were at the start of a period of assessment in the selection process for the Territorial Army's section of the SAS, after completing a period of training and preparation.
Rest in Peace lads.
