Donna Horn, of Wellingborough, Northamptonshire, was just 15 when the mistake occurred - she was left paralysed from the neck down.
The leukaemia treatment vincristine should have been injected into 18-year-old Wayne Jowett's vein.
Wayne had been unconscious since the incident last month.
He died in the hospital's intensive care unit on Friday morning. He was killed by a slow, creeping paralysis that eventually stopped his heart.
Wayne's death comes just 24 hours after an inquest into the death of a female patient who took eight years to die after being injected with vincristine in the spine.
Similar case
Donna Horn, of Wellingborough, Northamptonshire, was just 15 when the mistake occurred - she was left paralysed from the neck down.
The coroner at her case called for changes to syringe design to make another such tragedy impossible.
Vincristine, an effective treatment for leukaemia, carries a warning to say it must never be injected into the spine.
Vincristine is highly toxic, and can only be tolerated if injected slowly into a vein - if delivered into the spine, it causes devastating, and irreparable, nerve damage.
However, it is known that exactly the same mistake has been made in UK hospitals on 13 occasions over the last 15 years.
Wayne's family, from Keyworth in Nottinghamshire, have issued a statement saying they wish to left to grieve in peace.
Paul Balen, the family's solicitor, said close relatives had kept a bedside vigil since the blunder.
He said: "Mr Jowett died at around 8am today. Obviously the family, who are from Nottinghamshire, are very distressed and have asked to be left alone. They may feel able to talk about what has happened at some time in the future."
Mr Balen added: "My clients have been appalled to learn that so many other families have suffered as a result of similar mistakes."
Hospital response
The chief executive of the Queen's Medical Centre, John MacDonald, said the hospital had let the Jowett family down.
"We apologise unreservedly to the family and would like to express our deepest sympathy."
He added: "A serious mistake was made when Wayne's drug treatment was administered wrongly.
"In spite of immediate action being taken it proved impossible to save Wayne."
Mr MacDonald said staff had been reminded to follow strict protocols and procedures for administering such drugs to patients.
He said: "A full internal inquiry has already been started to discover what went wrong. And if there are any lessons to be learnt from this then they will be."
Death inevitable
At the time that the error was made public, Dr Kate Law, from the Cancer Research Campaign, said: "Sadly, in these cases, death is inevitable.
"The drug just destroys the nerves and it causes a massive inflammatory reaction."
Although hospitals have strict guidelines reinforcing the need for care when giving powerful chemotherapies, often the drugs are kept side by side in drug fridges.
In chemotherapy regimens for leukaemia, vincristine is often given alongside methotrexate, which is meant to be given into the spine.
The Department of Health set up a committee - chaired by Chief Medical Officer Professor Liam Donaldson - to come up with ways of reducing the number of medical mistakes within the NHS.
Its report in June 2000 urged the government to examine ways of completely wiping out deaths and disability caused by "wrongly administered spinal injections" by the year 2001.
The department told BBC News Online that no method for eradicating this problem had been discovered, and that the deadline had been put back to the end of this year.
A Department of Health spokesman said: "We are very sorry to hear of the tragic case of this young man.
"This is a rare and catastrophic event which has happened in this and other and other countries over the last 20 years.
"It is potentially avoidable and a major new initiative is being taken to try to address a problem which has not been solved by previous action."
The new initiative includes introducing a mandatory system for reporting medical mistakes.
Nottinghamshire Police today confirmed they had been called in to investigate the circumstances surrounding Mr Jowett's death.
A spokesman said: "We have been asked to investigate by the coroner and a report will be prepared."