George, now 47, was present in the dock of the Court of Appeal in London as the Lord Chief Justice, Lord Phillips, and two other senior judges allowed the appeal.
The court heard that George will now face a re-trial.
It was his second legal challenge to a jury's 2001 verdict that he shot the 37-year-old BBC News, Crimewatch and Holiday presenter on the steps of her home in Fulham, south west London, in April 1999.
The new appeal centred on fresh scientific evidence relating to a single microscopic speck of firearm discharge residue found in George's coat after his arrest a year after the shooting.
No application for bail was made today and Lord Phillips said that George would be remanded in custody.
George, dressed in a sober suit and dark blue shirt, smiled at his sister, Michelle Diskin, as he was led out of the dock of the packed courtroom.
Lord Phillips warned the media to be "extremely careful" not to publish anything that might prejudice the new trial.
George is to be rearraigned on a fresh indictment on the charge of murder within the next two months.
Earlier Lord Phillips announced: "The verdict is unsafe. The conviction will be quashed."
Jill Dando's fiance expressed disappointment at today's decision.
Alan Farthing, who is a consultant gynaecologist, said: "I am disappointed, particularly for those of us who have suffered the tragedy of losing a loved one prematurely.
"We all cope in our various ways and have had to move on with our lives.
"Therefore I have great sympathy for those who will now have to take emotional steps back in time to recall again, in detail, their involvement in this tragic case.
"I will be making no further public comment."
George was found guilty by a majority of 10 to one at his Old Bailey trial.
His first challenge to his conviction was rejected by the Court of Appeal in July 2002.
The latest proceedings follow a decision by the Criminal Cases Review Commission (CCRC), an independent body which investigates possible miscarriages of justice, to refer George's case back to the Court of Appeal.
George was present in the dock throughout the recent hearing before the Lord Chief Justice, Lord Phillips, Lord Justice Leveson and Mr Justice Simon.
Legal argument centred on fresh doubts about evidence relating to a microscopic speck of firearm discharge residue (FDR) found in the pocket lining of George's coat, which was seized after his arrest a year after the shooting.
George, who lived about half a mile from Miss Dando's home in Gowan Avenue, has always denied being her murderer.
His QC, William Clegg, told the appeal judges that the firearm discharge residue was "inconclusive" and could offer no assistance to a jury in assessing guilt.
Mr Clegg referred to a Forensic Science Service report which concluded last year that "it would be just as likely that a single particle of discharge residue would have been recovered from the pocket of Mr George's coat whether or not he was the person who shot Miss Dando nearly a year previously".
George's case is that too much weight was placed on the significance of that "neutral" evidence at trial.
Lord Phillips said today: "It is impossible to know what weight, if any, the jury attached to the FDR evidence.
"It is equally impossible to know what verdict they would have reached had they been told as we were told, by the witnesses who gave evidence before us, that it was just as likely that the single particle of FDR came from some extraneous source as it was that it came from a gun fired by the appellant."
Speaking outside the court, solicitor Jeremy Moore, from George's legal team, said: "We are very pleased with the outcome of today.
"This is only the latest hurdle in what has been a very long road and we now look forward very much to preparing for the trial.
"Barry is also very pleased. He is very pleased as well."
Reading out a summary of the court's decision, Lord Phillips said: "The prosecution called expert witnesses at the trial whose evidence suggested that it was likely that the particle of FDR came from a gun fired by Barry George rather than from some other source.
"Those witnesses and other witnesses from the Forensic Science Service told the Court of Appeal that this was not the right conclusion to draw from the discovery of the particle of FDR.
"It was, in fact, no more likely that the particle had come from a gun fired by Barry George than that it had come from some other source.
"The Court of Appeal concluded that, if this evidence had been given to the jury at the trial, there is no certainty that they would have found Barry George guilty.
"For this reason his conviction had to be quashed."
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