Cuomo said Judith Kaye, the state's former chief judge, would take over the investigation as a special prosecutor, The New York Times reported. Kaye now works for New York City law firm Skadden Arps.
Paterson, under fire for allegedly intervening in a domestic-violence case involving a top aide, announced in February he would not run for a full term. He became governor when Eliot Spitzer resigned in 2008 after his use of escorts became public.
Cuomo has also been investigating whether Paterson improperly sought free World Series tickets from the Yankees.
"An investigation of a sitting governor of New York is a complex weighty matter indeed," Cuomo told reporters in a conference call. "We have determined there are credible issues that need to be resolved."
Cuomo, son of former Gov. Mario Cuomo, is expected to announce his candidacy in the next few weeks. He had been expected to challenge Paterson in the Democratic primary.
Paterson is New York's first black governor, and some black politicians have criticized Cuomo's investigation.