The 70-year-old star, who rose to fame on the Scottish folk music scene, will continue to work in TV and on stage.
His spokeswoman said: "Billy Connolly recently underwent minor surgery in America after being diagnosed with the very early stages of prostate cancer. The operation was a total success, and Billy is fully recovered.
"In addition, Billy has been assessed as having the initial symptoms of Parkinson's disease, for which he is receiving the appropriate treatment.
"Billy has been assured by experts that the findings will in no way inhibit or affect his ability to work, and he will start filming a TV series in the near future, as well as undertaking an extensive theatrical tour of New Zealand in the new year."
The Glasgow-born star, affectionately known as the Big Yin, began his working life in the Clyde shipyards but soon moved into entertainment with folk singing and comedy performances.
A string of appearances on Michael Parkinson's chatshow made him a household name and helped launch a career that saw him perform sell-out stand-up shows around the world, present a series of documentaries and become an in-demand character actor.
He is married to New Zealand-born actress and psychologist Pamela Stephenson whose biography of her husband, simply called Billy, was a huge best-seller.
He is one of around 127,000 Britons with the disease, which is caused by a loss of brain cells that produce a chemical messenger called dopamine.
Symptoms differ from case to case but often include a tremor or fine shake while the person is at rest, rigidity of muscles, slowness of movement and unsteady balance.
Other possible symptoms can include memory loss and earlier this year, Connolly admitted he had started to forget his lines during performances.
Speaking about it, he said: "This is f*****g terrifying. I feel like I'm going out of my mind."
There is no cure for Parkinson's and scientists have been unable to work out why people get the condition.
Symptoms can be controlled using a combination of drugs, therapies and occasionally surgery, but often more care and support may be needed as they progress.
Actor Bob Hoskins announced his retirement last year after being diagnosed with the disease.
The disease was identified by - and named after - Dr James Parkinson who wrote An Essay On The Shaking Palsy in 1817 which established it as a recognised medical condition.
Connolly was made a CBE in the 2003 Queen's Birthday Honours and awarded the freedom of his home city in 2010.
Singer Bob Geldof said his "great friend" would not be deterred by the diagnosis.
He told Channel 5 News: "He's helped me lots in my endeavours. Pam and Bill are great mates. He's as strong as an ox mentally from everything he's been through as a kid. So I don't think this will deter him from being that individual that we know."
Glasgow City Council's Lord Provost Sadie Docherty has now sent a message to Connolly on behalf of the people of Glasgow. It said: "I read today of your recent health difficulties and was pleased to learn that you are on the mend following surgery. On behalf of the people of Glasgow, who welcomed you so warmly when you were made a Freeman of this city, I send our very best wishes and support to you and your family."
Connolly said he was very touched by the message.
In a response to the Lord Provost, he wrote: "I was very touched to receive a get well message from the Lady/Lord Provost. Please convey my best wishes to her."
Celtic football club has passed on its best wishes to life-long fan Connolly.
A statement on the club's website said: "Everyone at Celtic would like to pass on our thoughts and best wishes to Billy Connolly.
"Always a popular figure at Celtic Park, Billy is a comedy legend as well as being a star of television and films."
Former Celtic player Stiliyan Petrov, who recently battled acute leukaemia, tweeted: "@celticfc thoughts and prayers ate with you Billy."
Steve Ford, Chief Executive at Parkinson's UK, said: "Put simply Billy Connolly is a much loved comedy legend and we are sorry to hear that he is being treated for the early symptoms of Parkinson's. One person every hour will be diagnosed with Parkinson's in the UK, despite this it remains a little understood condition and we salute Billy's bravery in speaking out about his condition at this difficult time.
"There are 127,000 people in the UK, like Billy, living with Parkinson's. Parkinson's can be a very difficult condition to diagnose, as no two people with Parkinson's are the same, with symptoms - such a slowness of movement or tremor - changing on a daily, or even hourly basis.
"Many people, with the right medication, continue to live a full and active live with Parkinson's, but for some, it can be life changing and it is vital that Billy gets the support he needs to live with this complex condition.
"We wish Billy and his family all the best as they come to terms with this upsetting diagnosis."
Owen Sharp, chief executive of Prostate Cancer UK, said: "It is always concerning to hear that anyone is faced with prostate cancer, but it is very good news if Billy Connolly's treatment has been successful and we wish him the best in his recovery.
"Prostate cancer is the most common cancer in men, with 250,000 men living with the disease in the UK right now. In fact, a man dies from the disease every hour but it can often be treated successfully if caught early enough. That's why it is vital that more men and their families think about prostate health and take early action so more men survive and have a better quality of life in the future."