The jury sent State Supreme Court Justice Maxwell Wiley a note -- the third since April 29 -- saying it was unable to reach a unanimous decision on the guilt or innocence of bodega worker Pedro Hernandez.
On two previous occasions, including Tuesday and Friday, the judge ordered the jury to keep deliberating.
A court hearing has been set for June 10 to set a date for a new trial.
One member of the jury of seven men and five women -- who identified himself only as Adam, juror No. 11 -- told reporters he held out against conviction.
"Ultimately, I couldn't find enough evidence that wasn't circumstantial to convict," he said. "I couldn't get there."
The juror described the deliberations as respectful but sometimes heated.
A man identified as juror No. 3 urged prosecutors to retry the case. Juror No. 1, Alian Pahhan, shouted to reporters as she left a news conference.
"Nothing's impossible," she said. "They'll get him next time ... Pedro Hernandez, you know what you did."
'Haunted by demons"
Stanley Patz, father of the victim, said he remains sure of Hernandez's guilt.
"I'm so convinced that Pedro Hernandez kidnapped and killed my son," he told reporters. "He's a guilty man who's been conscience-stricken due to his deeds and haunted by demons ever since that day."
Defense attorney Harvey Fishbein said he was disappointed there was a hung jury but was prepared to defend his client in the event of another trial.
"Pedro's reaction is a very simple one: 'Does that mean I don't have to come back on Monday?'" Fishbein said. "Pedro is not a keen and cunning individual. Pedro is not a planner. Pedro is not someone who would spend 35 years avoiding criminal prosecution. Pedro is not the person that should be on trial in this case."
In a statement, Manhattan District Attorney Cyrus Vance Jr. said there is "clear and corroborated evidence of the defendant's guilt beyond a reasonable doubt."
"The challenges in this case were exacerbated by the passage of time, but they should not, and did not, deter us," he said.
Vance also thanked Patz's family for the "courage and determination they have shown over the past 36 years."
"The legacy they have built in the four decades since this tragedy occurred, both in raising awareness about the plight of missing children and through the creation of laws to protect them, has made our city, and our society, safer for children."
'This man did it'
Hernandez confessed to police three years ago, but his lawyers said he made up his account of the crime.
"We have now seen the confession," Stanley Patz said. "My family has seen the confession many times. So it's no longer as shocking but it's still just chilling. This man did it. He said it. How many times does a man have to confess before someone believes him?"
Etan's parents have waited more than 35 years for justice, but some have questioned whether that is even possible in Hernandez's case. His lawyer has said he is mentally challenged, severely mentally ill and unable to tell whether he committed the crime.
Hernandez told police in a taped statement that he lured Etan into a basement as the boy was on his way to a bus stop in Lower Manhattan. He said he killed the boy and threw his body away in a plastic bag.
Severe mental illness?
Neither the child nor his remains have been recovered.
Hernandez has been diagnosed with schizotypal personality disorder, one of a group of conditions informally thought of as "eccentric" personality disorders. He has an "IQ in the borderline-to-mild mental retardation range," Fishbein has said
Police interrogated Hernandez for 7½ hours before he confessed.
"I think anyone who sees these confessions will understand that when the police were finished, Mr. Hernandez believed he had killed Etan. But that doesn't mean he actually did, and that's the whole point of this case," Fishbein has said.
In November, a New York judge ruled that Hernandez's confession and his waiving of his Miranda rights were legal, making the confession admissible in court.
Hernandez is charged with two counts of second-degree murder for allegedly intending to cause the boy's death and for a killing that occurred during a kidnapping.
Another man's name has also hung over the Etan Patz case for years: Jose Antonio Ramos, a convicted child molester acquainted with Etan's babysitter. Etan's parents, Stan and Julia Patz, sued Ramos in 2001. The boy was officially declared dead as part of that lawsuit.
A judge found Ramos responsible for the boy's death and ordered him to pay the family $2 million, money the Patz family has not received.
Though Ramos was at the center of investigations for years, he has never been charged. He served a 20-year prison sentence in Pennsylvania for molesting another boy and was set to be released in 2012.
He was immediately rearrested
upon leaving jail in 2012 on charges of failing to register as a sex offender, The Associated Press reported.
Since their young son's disappearance, the Patzes have worked to keep the case alive and to create awareness of missing children in the United States.
In the early 1980s, Etan's photo appeared on milk cartons across the country, and news media focused on the search for him and other missing children.
"It awakened America," said Ernie Allen
, president and chief executive officer of the National Center for Missing & Exploited Children
. "It was the beginning of a missing children's movement."
The number of children who are kidnapped and killed has remained steady -- it has always been a relatively small number
-- but awareness of the cases has skyrocketed, experts said.
The news industry was expanding to cable television, and sweet images of children appeared along with distraught parents begging for their safe return. The fear rising across the nation sparked awareness and prompted change from politicians and police.
In 1984, Congress passed the Missing Children's Assistance Act, which led to the creation of the National Center for Missing & Exploited Children.
President Ronald Reagan opened the center in a White House ceremony in 1984. It soon began operating a 24-hour toll-free hot line on which callers could report information about missing boys and girls.