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He has been ordered instead to pay a £1,000 fine and has been put on probation for three years. An order to pay 250 guineas in costs still stands from his earlier hearing.
Jones, aged 25, has also promised to continue medical treatment with a psychiatrist.
On 30 October the guitarist pleaded guilty to two drugs offences, possession of cannabis and permitting his home to be used by others smoking cannabis.
Catapulted to fame
The judge handed down a nine-month jail term - but Jones was released on bail a day later pending his appeal.
The drugs - enough to make up to 10 reefer cigarettes - were found during a police raid of his Kensington flat in May after a party.
James Comyn QC told the Appeal court his client had since "cut out drugs, soft, hard or what you will for the future".
He said Jones was a highly intelligent and extremely sensitive young man who had been catapulted to fame.
He added: "It may sound trite but Jones has suffered every single day since sentence was passed upon him - a suffering which cannot be removed and may be regarded as penalty enough.
"This man is at the very crossroads of life and if you uphold the sentence it is liable to break him and his career.
"Brian Jones is at your mercy and it is mercy that he seeks."
Dr Anthony Flood, a consultant psychiatrist who treated Jones for three weeks while he was on bail, also spoke on his behalf.
He said waiting for the appeal had had "an astounding effect" on the young musician.
"I think if one put a reefer within half a mile of Brian Jones he would start running," Dr Flood said.
The court also heard from Dr Leonard Henry, a Harley Street psychiatrist, who described Jones as emotionally unstable.
The judges' view was given by Lord Parker who said it was difficult to understand the kind of life Jones led.
He said there was nothing wrong with the jail term imposed on Jones but the court had decided "not without considerable doubt" a degree of mercy would be shown.
The fine of £1,000 was the maximum allowed under the law.
Stones' lead singer Mick Jagger was in court to hear the judges' decision. He and fellow band member Keith Richards had been in court earlier in the year following a different drugs raid. They had also had their jail terms overturned.
As Jones left the court, he said: "I am very happy to have my freedom."
Brian Jones drove straight from the Appeal Court to the dentist to have treatment for a toothache.
In 1969 he split with the Rolling Stones quoting musical differences with the rest of the band.
He wanted to pursue his love for rhythmn 'n' blues. It was at this time he also bought the 15th century Sussex farmhouse where AA Milne is said to have written Winnie the Pooh.
In July 1969 Jones was found dead in the swimming pool of his home near Hartfield.
A coroner recorded a verdict of death by misadventure. He had died from drowning under the influence of alcohol and drugs.
At his funeral, Canon Evan Hopkins read a telegramme Brian Jones had sent to his parents during his drugs trial. It said: "Don't judge me too harshly."
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