Stephens said in the live video that he has killed more than a dozen people, police said, though they have announced no other victims. The two men did not know each other, police said.
Stills from a Facebook video shows the moment that Stephens shot and killed a man.
Steve Stephens is alleged to have posted horrific footage online which showed him getting out of his auto and executing the victim in the street at random.
This story has been corrected to show that Facebook now says the killing was recorded and uploaded to Facebook, not broadcast live.
Cleveland police chief Calvin Williams confirmed one person had been killed, but said police did not know of any other victims.
Police have warned residents of the surrounding states of Pennsylvania, New York, Indiana and MI to be on the lookout for Stephens, who they said may have travelled out of Ohio.
In a statement, a Facebook spokesperson said the shooting was a "horrific crime".
Stephens is an employee at Beech Brook, a children's behavioral health center in OH, a company spokesperson told ABC News.
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A spokeswoman for Beech Brook, Stephens' employer, said they are "shocked and horrified".
The video remained up on Facebook for almost three hours before it was taken down. He tells the man to say a woman's name and pulls out his gun.
A series of posts Sunday on what authorities said was Stephens' Facebook page complained that he had "lost everything" to gambling.
The video shows a man pulling over in a auto as he says 'find me somebody to kill, gonna kill this guy right here.
Cleveland Mayor Frank Jackson publicly urged Stephens to turn himself into police and not to "do anymore harm to anybody". The video was posted Sunday afternoon and Facebook later disabled access to it.
Stephens is a black male who is 6-foot-1 and weighs 244 pounds.
Distraught relatives of Robert Godwin Snr, 74, described the elderly victim as "a good guy", hours after a gunman, suspected to be Steve Stephens, 37, shot him dead. Police initially said the victim's last name was spelled Goodwin. The posts named specific people whom the user wanted to talk to, and at one point he wrote "I killed 12 people today" and wouldn't stop until he could speak to his mother and a second woman. Anyone with information is urged to call 911. Authorities believe that he's driving a white Ford Fusion with the temporary license plate number E363630, and have warned anybody who sees him to call 9-11, rather than attempt to confront him.
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