Kasparov, a fierce critic of Russian President Vladimir Putin, had hoped to persuade the court that putting Ablyazov in Moscow's hands would amount to a violation of his human rights.
Ablyazov, a former minister and banker turned fugitive, is wanted by Russia, Ukraine and Kazakhstan on embezzlement charges which he says were trumped up by his arch-foe, the president of his oil-rich nation, Nursultan Nazarbayev.
He was arrested in France in July last year after fleeing London where he was also the subject of several civil claims in connection with the case.
After deliberation, Judge Luc Fontaine said he would not "proceed with the hearing of witnesses" as this was not allowed in extradition cases.
However he did not rule out the possibility entirely, saying the witnesses could be called at a later stage if further information was required.
Lawyer Jean-Pierre Mignard said he would appeal the decision.
"This is unacceptable, I regret the bad image given by the French justice system."
Kasparov said at a press conference yesterday that while he "was not interested" in the details of the case, he wanted to offer testimony on the likely fate that would await Ablyazov should he be extradited to Russia.
Ablyazov was once a member of the inner circle of the Kazakhstan elite and served as energy and trade minister before he fell from grace and stirred up opposition to the country's iron-fisted government.
He is accused of siphoning off billions of dollars in state and investors' funds from the BTA bank, which also had interests in Ukraine and Russia -- both of whom have demanded his extradition.