The Walking With The Wounded charity said that five days into the gruelling trek, "it became obvious that underneath the concrete determination of all the team members, the harsh reality of the Antarctic was starting to take its toll".
The three teams, made up of wounded veterans from Britain, the United States and the Commonwealth (represented by Canada and Australia), will trek the final 112 kilometres together and aim to arrive by next Friday or Saturday, the charity said.
The charity's expedition director Ed Parker said the teams had been progressing well but the "unprecedented terrain" had placed a lot of stress on the wounded veterans, who include seven amputees.
"With careful consultation from our doctor and race team, we have put the race on hold," Parker said.
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"This does not mean that the expedition is over. Far from it. We came down here, determined to get 12 men and women, all injured in conflict, to the South Pole, and so we will."
The veterans are enduring temperatures as low as minus 35 degrees Celsius and wind speeds of around 80 kilometres per hour.
They are pulling 70-kilogram sleds throughout the 335-kilometre race.
Walking With The Wounded raises funds to retrain injured troops and help them find new careers outside the military.
Launching the race at London's Trafalgar Square last month, Harry described the trek as "a wonderful display of courage" by the troops.
Harry joined the charity for part of a trek to the North Pole in 2011, but had to come home early to be best man at his brother Prince William's wedding.