The small plane was setting off on a mission over the Mediterranean around 07:20 local time when it nose-dived and disintegrated in a ball of flames.
There was no explosion prior to the crash, according to eyewitnesses.
According to Maltese officials, the flight had been part of a French operation to track illicit trafficking of people and drugs.
An investigation into the causes of the crash is to be conducted by the French defense ministry.
The airport has re-opened with some disruption expected to the day’s scheduled flights.
French Defense Minister Jean-Yves Le Drian confirmed that three of the victims were ministry staff while the other two were contractors.
Early reports that the victims had been officials from the EU’s border agency were quickly denied. Frontex said none of its staff had been involved.
Malta said the flight had been due to return to the island within hours without landing in third countries.
Eyewitnesses told the Times of Malta the plane had taken off before turning on its side and going “straight down into the ground”.
Video shot from another plane at the airport shows flames and thick black smoke billowing from the site of the crash, next to the runway.
All of the victims’ remains have been found, the Maltese government said.
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