On June 25 an appeals court had overturned an earlier ruling by the same Vienna criminal court in favour of Aslan Gagiev's extradition, citing concerns about his human rights if sent to Russia.
But the criminal court's presiding judge Michael Spinn ruled on Wednesday that there were no "serious grounds" not to extradite Gagiev and that it was "not at all proven" that he faced torture and maltreatment in Russia.
The appeals court had also called on Russia's public prosecutor to provide assurances that Gagiev's human rights would not be infringed. This has since been given to the Austrian authorities.
Georgian-born Gagiev, 45, who was arrested in Vienna in January, is accused by Russian authorities of personally carrying out at least six murders.
His gang's alleged victims include the deputy vice-president of the Russian republic of North Ossetia, the deputy public prosecutor, a mayor and the chief of police, Austrian police said at the time of his arrest.
The latest ruling can again be appealed.