Prince Albert's battle to overcome his stammering he endured since his childhood was captured in the 2010 British historic film.
It depicts how Australian speech and language therapist Lionel Logue played by Geoffrey Rush helped him to make public speeches as he takes the throne after his brother abdicated and Britain prepared for war.
But the new discovery could replace speech and language therapy currently offered by the NHS.
The discovery was made by chance after a 61-year-old man who was downing up to three litres of wine a day found his had gone - after he began taking baclofen.
Baclofen is a muscle relaxant and widely used for the treatment of stiff muscles caused by multiple sclerosis and spinal cord diseases.
Stammering or stuttering is a common condition beginning in childhood with around one in 20 young children go through a phase of stammering.
Although four in five children grow out of it, the NHS estimated stammering affects around one in 100 adults, with men four times more likely to stammer than women.
Psychologist Dr Esther Beraha, of the University of Amsterdam, said the findings would need to be confirmed in much larger clinical studies including participants who are not alcohol dependent.
But she said there are potentially plausible biological explanations, one of which is that muscle tension is a factor in stuttering.
So the properties of baclofen could be acting on the respiratory muscles and those in the neck and face.
Secondly, some studies suggest baclofen reduces anxiety in people who are alcohol dependent.
Stress is also associated with stuttering.
Another possibility is baclofen may indirectly reduce production of the neurotransmitter dopamine, higher levels of which are associated with the speech impediment.
It has recently been used to treat alcohol dependence because it is thought to target the nerve centres in the brain involved in reward and addiction.
As yet, the evidence for baclofen's impact on curbing alcohol craving and improving abstinence has been mixed so has not yet been officially licensed for alcoholism.
But it seemed to work for the patient who regularly drank two to three litres of wine every day and admitted to having had a problematic relationship with alcohol for 20 years.
As well as sleep problems and a history of depression, he also stuttered in Dutch, his second language.
The patient said: "Personally, as mentioned in the report, I think my stuttering had a lot to do with finding the right words when speaking Dutch.
"Also, anxiety and nervousness played a key role, especially on my first few weeks visiting my physician.
"Detoxification was also the cause for anxiety and nervousness, which slowly ceased as my baclofen dosage increased.
"At present I do not have any problems with stuttering.
Dr Beraha said: "The patient reported to be socially isolated, unable to work and
having lost several relationships."
The potential impact of baclofen on stuttering came to light when the man agreed to take part in a clinical trial looking at treating alcohol dependence with the drug.
As part of the trial he ended up taking 120 mg of baclofen every day for 10 weeks.
But once a daily dose of 90 mg had been reached, his doctors noticed that he had stopped stuttering.
Nevertheless, the man complained of sleepiness, stiff muscles and heavy legs on this dose, prompting his doctors to gradually taper down the dose to zero, say medics writing in BMJ Case Reports.
However, once he had stopped taking baclofen, he returned to his former level of drinking and his stutter re-emerged.
He was therefore advised to continue taking the drug at a daily dose of 90 mg after which he stopped drinking for a prolonged period and his stutter disappeared.
Dr Beraha and colleagues said the drug may offer a new treatment option for stuttering.
She said: "This case illustrates the potential efficacy of a high-dose baclofen treatment for patients with alcohol dependency."
Added Dr Beraha: "Based on this case study, baclofen might have beneficial effects on stuttering either through muscle relaxation, reduction of anxiety, or indirect dopaminergic inhibition.
"Further research is warranted using larger samples and placebo-controlled studies."