The seven-year-old girl, whose mother drank excessively while pregnant, was born with severe brain damage.
The Court of Appeal ruled in December that the girl, now in care, was not legally entitled to compensation.
Permission to appeal was refused because an arguable point of law was not raised, the Supreme Court said.
An application was made to the Criminal Injuries Compensation Authority on her behalf on the basis that the girl was the victim of crime because her mother had administered a noxious substance to her.
The case was originally brought by a council in north-west England, which cannot be named for legal reasons. The Court of Appeal rejected a decision of the First Tier Tribunal awarding her compensation.
Foetal alcohol syndrome
Heavy drinking during pregnancy can lead to foetal alcohol syndrome (FAS) when alcohol passes across the placenta from the mother to the developing foetus.
The foetus cannot process alcohol effectively until the liver is fully developed and the high levels of alcohol can affect the development of organs and the brain.
Children with FAS are born with a range of disabilities, are often shorter than average and some have learning and behavioural difficulties.
People with the syndrome can have differences in their facial features such as a flat nose bridge, a small head and a thin upper lip.
It is thought that foetuses are most at risk during the first three months of pregnancy when organs are forming - but damage can occur at any time.
The NHS recommends that pregnant women should not drink at all - adding that those who choose to have a drink should have no more than two units of alcohol once or twice a week.

Nicknamed Psycho, the former Nottingham Forest defender played in a World Cup semi-final in 1990 but has signed for a team dubbed "the worst in the UK".
Longford AFC, based in Gloucestershire, have lost all 18 games this season, scoring one goal and letting in 179.
Pearce said the grassroots game was "essential" to the sport's lifeblood.
"I started out in non-league football playing for Wealdstone before joining Coventry City," he said.
"So I'm looking forward to working with manager Nick Dawe, and helping a group of talented young footballers achieve their dreams and, hopefully transforming both the dressing room confidence, as well as the on-the-pitch performance, and seeing how far they can go up the league."
179
goals conceded
0-17 Loss at home to Bibury
16-0 Loss away to Lydney Town Reserves
0-15 Loss at home to Quedgeley Wanderers
14-0 Loss away to Woolaston
Longford AFC are rooted to the bottom of the Gloucestershire Northern Senior League Division Two.
With an awful record and a growing interest from the media, the side has been dubbed "the worst team in England".
But that could change following the recruitment of their star January signing, who is expected to train with the squad before playing one game with them.
Nick Dawe, Longford AFC manager, said the opportunity would give the side "the best chance possible to go further than we ever have done before".
"A year ago we were playing in front of one man and his dog and now we are set to [attract] hundreds for Stuart Pearce's debut," he added.
"Stuart knows that there will be no special treatment, he will still need to pay his Â£5 a game subs."
Pearce, a winner of 78 England caps, was part of Sir Bobby Robson's squad that reached the last four of the World Cup in Italy in 1990, and the Terry Venables-led side that was eliminated at the same stage of Euro '96.
He retired from playing six years after that, having enjoyed spells at Newcastle United, Manchester City, West Ham, Coventry and non-league Wealdstone as well as captaining Forest under the management of Brian Clough.
A child born with foetal alcohol syndrome has been refused permission to take her case for criminal injuries compensation to the UK Supreme Court.
[NEXT_CONCEPT]
England football legend Stuart Pearce is set to come out of retirement aged 53 in a bid to transform the fortunes of a struggling non-league side.