Mr Straw said the decision was based on medical evidence that Biggs's condition had deteriorated and he was not expected to recover.
Biggs, who turns 80 this weekend, is severely ill in hospital with pneumonia and doctors have said there is "not much hope".
Last month the Justice Secretary rejected Biggs's application for parole on the grounds that the robber was "wholly unrepentant" about his crimes.
But the decision on compassionate release was based on "different considerations", Mr Straw said.
"The medical evidence clearly shows that Mr Biggs is very ill and that his condition has deteriorated recently, culminating in his re-admission to hospital. His condition is not expected to improve," he said.
"It is for that reason that I am granting Mr Biggs compassionate release on medical grounds.
"I have therefore been satisfied that the relevant conditions have been met, which I was not in respect of the recommendation for parole."
Biggs was rushed to the Norwich and Norfolk Hospital from his cell at Norwich prison on Tuesday.
At the same time, his son, Michael, made a desperate plea for his father's release.
The decision means Biggs can celebrate his birthday on Saturday a free man. It will be 46 years to the day since the robbery.
Although the decision was made today, Biggs will remain on bed watch overnight.
The three Prison Service staff watching him will be withdrawn tomorrow, once the licence for his release is finalised.
His condition means he will be unable to celebrate his release.
After a series of strokes, he is bedridden, fed through a tube and barely able to communicate.
Biggs, from Lambeth, south London, was a member of a 15-strong gang which attacked the Glasgow to London mail train at Ledburn, Buckinghamshire, in August 1963, and made off with £2.6 million in used banknotes.
He was given a 30-year sentence but after 15 months he escaped from Wandsworth prison in south west London by climbing a 30ft wall and fleeing in a furniture van.
Biggs was on the run for more than 30 years, living in Australia and Brazil before returning to the UK voluntarily in 2001 in search of medical treatment.
He was locked up in Belmarsh high-security prison on his return before being moved to a specialist medical unit at Norwich prison.
Judy Totton, a spokeswoman for Michael Biggs, said: "He is absolutely delighted and he hopes that his father will survive long enough to see his 80th birthday on Saturday."
Biggs's legal adviser, Giovanni Di Stefano, told Sky News: "He is being released effectively to die and that cannot be considered a victory. But it's a victory for common sense and Mr Straw has made the right decision."
He went on: "This man is ill, he's going to die, he is not going to any pub or going to Rio, he is going to stay in hospital."
Mr Di Stefano thanked the Justice Secretary for his decision.
Similar pleas for release on compassionate grounds were made in 2006, 2007 and in June, but refused.
Prison service staff presented the Justice Secretary's decision letter to Biggs at the hospital just after 5.30pm.
Harry Fletcher, assistant general secretary of Napo, the probation officers' union welcomed the decision, but he said Mr Straw should have released the Great Train Robber in July when it was recommended by the Parole Board.
He said: "I welcome this decision. His medical condition was pretty desperate two months ago - he couldn't walk, he could barely talk and he posed no risk to anybody.
"It's just a great shame he wasn't released two months ago on the recommendation of the Parole Board.
"There have been just eleven occasions in the last five years when Parole Board decisions involving men serving 15 years or more have been overturned."
"This year it has happened on just one occasion - that of Ronnie Biggs."
The judicial review of Mr Straw's Parole Board decision will be dropped, Mr Di Stefano confirmed.
If his condition was to improve, Biggs would be transferred to a nursing home in Barnet, north London, near his son's home.
Mr Di Stefano said: "He is going to stay in hospital. He is seriously, seriously, critically ill, and that's the basis that Mr Straw has released him and we thank him for making that decision.
"In the event he is released from hospital, he will go to a nursing home in Barnet, where he will stay for the rest of his days."
The train drivers' union Aslef criticised the decision.
General secretary Keith Norman said: "It's ludicrous that a man who was part of a gang that committed a violent crime and attacked an innocent man and hit him with an iron bar should be a person who deserves clemency."
Juliet Lyon, director of the Prison Reform Trust, said: "Given that he has suffered several strokes and is now bedridden, unable to communicate and fed through a tube, there is a strange irony in Jack Straw having formally to warn Ronnie Biggs that he must not breach strict licence conditions, including to behave well and not travel abroad nor offend again, or he will be liable to immediate recall to prison.
"Our prisons are filling up with infirm, elderly people for whom that bleak environment serves as a double punishment.
"The Justice Secretary will now need to give thought to the hundreds of prisoners over the age of 70, who, if they are still considered a danger to the public, should be held in secure homes for the elderly rather than in prisons which, day in, day out, run in breach of the Disability Discrimination Act.
"Prison is no place to grow old and die if there is any way that can be avoided."
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