The discovery could lead to a screening programme for the disorder - with sufferers given oxygen masks, or mouthguards that keep their airways open in bed.
Obstructive sleep apnoea (OSA) affects around 2.5 million Britons, and is most common among the elderly.
Sufferers snore loudly, then stop breathing for a few seconds, then to start snoring again with a start.
The oxygen deprivation can cause low blood pressure - and the body’s catching up effort strains the cardiovascular system.
Senior study author Professor Ricardo Osorio said his findings follow recent evidence of a link between decreased REM (rapid eye movement) sleep and dementia. This is the dreaming phase, and the more there is, the more restful the sleep.
He said: “Our data support testing whether clinical interventions aimed at OSA, such as treatment with CPAP or dental appliances, could be implemented during the early phase in which tissue damage precedes clinical symptoms and neuronal dysfunction, to mitigate the progression of cognitive impairment.”
A CPAP (continuous positive airway pressure) machine combats sleep apnoea by keeping the airways open, and forcing air in.
The study published in the American Journal of Respiratory and Critical Care Medicine identified biomarkers for amyloid beta proteins in participants with OSA. These clump together in the brains of Alzheimer’s patients.
What is more, they increased over time in proportion to snoring severity, meaning those with more sleep disturbances per hour had greater accumulation of amyloid over time.
Psychiatrist Prof Osorio, of New York University, said: “Several studies have suggested sleep disturbances might contribute to amyloid deposits and accelerate cognitive decline in those at risk for Alzheimer’s disease (AD).
“However, so far it has been challenging to verify causality for these associations because OSA and AD share risk factors and commonly coexist.”
In the study of 208 mentally healthy people, aged 55 to 90, his researchers performed lumbar punctures to analyse the amount of amyloid in their cerebrospinal fluid (CSF)
They then used PET (positron emission tomography) scans to measure deposits of the rogue proteins directly in the brain in a subset of the volunteers.
None of the participants was referred by a sleep centre, used CPAP to treat sleep apnoea, was depressed or had a condition that might affect their brain function.
But more than half were found to have sleep apnoea - 36.5 percent with a mild form and 16.8 percent whose condition was moderate to severe.
Exactly half the original group, 104, were then followed for two years which found a link between OSA severity and an increase in amyloid deposits in the brain. This was shown by analysis of cerebrospinal fluid, and confirmed by PET scans.
Surprisingly, it did not predict cognitive deterioration in these healthy elderly adult, suggesting these changes were occurring in the preclinical stages of Alzheimer’s.
Study co author Dr Andrew Varga, a sleep expert and neurologist at the Icahn School of Medicine at Mount Sinai in New York, said: “The relationship between amyloid burden and cognition is probably nonlinear and dependent on additional factors.”
The researchers believe it may also be attributable to the study’s relatively short duration, highly educated participants and use of tests that fail to discern changes in cognitive abilities that are subtle or sleep dependent.
Prof Osorio said: “Recent literature in both mice and humans suggests disturbed sleep leads to higher levels of brain soluble beta amyloid peptides, which aggregates to form senile plaques, a hallmark of Alzheimer’s disease.
“This pathological process might be present prior to cognitive decline, indicating that disturbed sleep can be both a consequence and a risk factor for Alzheimer’s disease.
“This longitudinal study shows that obstructive sleep apnea, very common in the elderly, can be a risk factor for developing Alzheimer’s disease.”
He added: “Results from this study, and the growing literature suggesting that OSA, cognitive decline and AD are related, may mean that age tips the known consequences of OSA from sleepiness, cardiovascular and metabolic dysfunction to brain impairment.
“If this is the case, then the potential benefit of developing better screening tools to diagnose OSA in the elderly who are often asymptomatic is enormous.”
Earlier this year another US team that pooled data from 14 studies showed people with sleep apnoea were 26 per cent more likely to develop dementia.
The number of tests in England to diagnose people with sleep disorders has more than doubled in a decade - from 69,919 in 2007 to 147,610 last year.
Experts said the rise may be down to more awareness about the serious health risks of sleep apnoea, which include increased chances of heart disease and stroke.
In the US, sleep apnoea affects 18 million people, or one in 15 Americans, according to the American Sleep Association.