Arafat died at the age of 75 in Percy military hospital near Paris on November 8, 2004. The claim of poisoning was made by his wife Suha Arafat.
According to her he was poisoned, possibly by highly radioactive polonium. Tests carried out in Switzerland apparently backed her claim of poisoning.
"The polonium poisoning had not been demonstrated and that they would not continue their inquiries," said a statement issued by the prosecutors in Nanterre.
He was diagnosed with a serious blood disorder and died of a stroke. Post-mortem examination not was carried out by authorities as his wife did not ask for it.
The issue of being poisoned was raised in 2012 when an investigation by al-Jazeera TV, in conjunction with Swiss analysts in Lausanne, found abnormal levels of polonium-210 on his personal effects.
Three teams of French, Swiss and Russian investigators took take samples from Arafat's tomb in Ramallah after his wife called for her late husband's body to be exhumed
But, earlier this year, one French prosecutor said the polonium samples were of an environmental nature.
The head of the Palestinian Authority's inquiry committee on Wednesday refused to accept the judges' conclusions.
Most Palestinians believe that Israel poisoned their late leader when he was put under house arrest but Tel Aviv has repeatedly denied any wrongdoing.