The crash occurred late Monday during an operation against jihadists in the Liptako region, near the borders with Burkina Faso and Niger.
The accident brought to 41 the number of French troops killed in the Sahel region since Paris intervened against Al-Qaeda-affiliated jihadists in Mali in 2013.
Three helicopters and a squadron of Mirage jets had arrived on Monday to support ground troops pursuing Islamist extremists.
Shortly after troops engaged the insurgents, who fled on motorbikes and in a pickup truck, a Tiger attack helicopter collided with a Cougar military transport helicopter, killing all 13 onboard the two aircraft.
Two black boxes found
Defence Ministry Florence Parly said Monday's operation was particularly risky because it took place in "total darkness" on a moonless night, but the exact cause of the collision remains unknown.
"The two black boxes from the helicopters have been recovered, they will be handed over to the relevant authorities to be analysed," the spokesman, Colonel Frederic Barbry, told BFMTV.
Barbry said no theory as to the cause of the crash is being ruled out, and it would be up to the investigation team to determine why the collision occurred.
"The pilots work with night vision binoculars that intensify the residual light when there is no moon, no sources of artificial light like in cities, as is the case in this region."
Barbry said the soldiers' bodies will be repatriated to the French mainland shortly.
Evening tribute to soldiers in Pau
In the southwestern French town of Pau, several hundred people including army veterans, serving officers, officials and ordinary citizens gathered on Tuesday night to pay tribute to the men.
Outside the town hall where flags were at half-mast, the crowd held a minute's silence before portraits of seven of the soldiers who were from a combat helicopter unit based in the city.
One of the victims was the son of French senator Jean-Marie Bockel, a centrist former government minister who sits on the senate's armed forces committee, the father confirmed to AFP.
Macron to lead ceremony at Invalides
President Emmanuel Macron would lead a commemorative service for the soldiers at the Invalides military hospital and museum in Paris on Monday.
"These 13 heroes had just one goal: To protect us. I bow my head in front of the pain of their families and comrades," Macron said on Twitter.
Mali's President Ibrahim Boubacar Keita said the soldiers had died for Mali and all Sahel countries, as well as for France.
"The loss is heavy but the peoples of the Sahel share your bereavement," he said in a letter to Macron.
