In a statement published on Wednesday, the Automobile Club de l'Ouest said the decision was made "in view of the changing health situation related to the coronavirus and the latest Government guidelines".
It continued: "Restrictions are currently in force in France and other countries around the world to curb the spread of coronavirus. Consequently, race organiser the Automobile Club de l'Ouest has decided, in conjunction with the Federation Internationale Automobile (FIA) and the World Endurance Championship (FIA WEC), to postpone the 24 Hours of Le Mans originally set for 13-14 June 2020."
The race was run during September in 1968, due to protests, strikes and civil unrest in France during that summer.
Pierre Fillon, President of the Automobile Club de l'Ouest, said: "Postponing the 24 Hours of Le Mans from the original dates in June is now the most appropriate way forward in the current exceptional circumstances.
"First and foremost, I urge everyone to avoid putting themselves, their loved ones and others at risk. The most important thing today is to curtail the spread of this virus. Our thoughts go out to medical staff working relentlessly for the sake of us all.
"The postponement of the 24 Hours of Le Mans means making changes to the WEC and ELMS calendars and we shall announce the new dates shortly. The safety and quality of our events will not be compromised.
"Competitors, sponsors, fans, media, medical services and organisers – it is time to pull together, more than ever before."
Gerard Neveu, CEO of FIA WEC and ELMS, added: "It is the right decision to delay in light of the current situation. We are now working on revising our WEC and ELMS calendars for the remainder of this season, and for the WEC's Season 9 which was due to start in September 2020.
"We will come back to you in the coming days with more news for both the FIA World Endurance Championship and the European Le Mans Series."
Moving Le Mans into the late summer will require a wholesale rejig of the 2020/21 WEC calendar.
The Silverstone opener was to be followed by a race at Monza on October 4 before the first non-European race at Fuji on November 1.
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