"After a brief pursuit, Stephens shot and killed himself", the police said on Twitter, providing no further details.
Stephens, who had no prior criminal record, was not suspected in any other killings, Cleveland officials had said.
The video in which police say he killed Godwin was taken down by Facebook after a few hours. A reward of $50,000 was offered for information regarding Stephens' location.
"We urge everyone to remain vigilant at all times, but we have no indications that the Cleveland murder suspect is in or headed to the Baltimore area", Smith said.
The shooting marked the latest video clip of a violent crime to turn up on Facebook, raising questions about how the world's biggest social media network moderates content.
Chief Williams: This is now a national search for Steve.
It wasn't immediately clear where Stephens had been since Sunday. Godwin did not seem to recognize the name.
Stephens was employed at Beech Brook, a behavioral health agency in northeastern OH that serves children, teenagers and families, according to a spokeswoman for the facility.
Police say the 37-year-old Stephens had a permit to carry a concealed weapon.
In the video, Stephens named a woman and tells the victim, Robert Godwin Sr., that "she's the reason that this is about to happen to you".
Owner Blake Frederick says suspect Steve Stephens visited Sherwin Shooting Sports within the past year.
"I want him to know what he took from us".
Facebook disabled Stephen's account within 23 minutes of receiving the first report about the video and fatal shooting. "I am sorry that all of this has happened". "We received reports about the third video, containing the man's live confession, only after it had ended".
Authorities had expanded the search nationwide for Stephens who was believed to be behind the random killing of an elderly man on Easter Sunday. Detectives were able to get in contact with him, telling him to surrender.
A man who identified himself as Stevie Steve is seen in a combination of stills from a video he broadcast of himself on Facebook in Cleveland, Ohio, April 16, 2017. Police in Philadelphia said eight elementary schools and a high school were locked down Monday while they investigated reported sightings but found nothing.
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