The Irish football association (FAI) appealed to FIFA for a replay after Thierry Henry handled the ball when setting up William Gallas's extra-time equaliser in Wednesday's 1-1 draw in Paris that sent France to the World Cup with a 2-1 aggregate win.
However, as was widely expected, FIFA wasted little time in rejecting the plea.
"FIFA has replied to the request made by the FAI to replay the World Cup play-off match between France and the Republic of Ireland," the body said on its website on Friday.
"In the reply, FIFA states that the result of the match cannot be changed and the match cannot be replayed. As is clearly mentioned in the Laws of the Game, during matches, decisions are taken by the referee and these decisions are final."
The FAI said on Thursday that the incident, which dominated news bulletins in Ireland all day as well as being discussed in parliament, had "damaged the integrity of the sport."
The Irish government supported the call for a replay and Prime Minister Brian Cowen said he would discuss the matter with French President Nicolas Sarkozy in Brussels.
However, Ireland manager Giovanni Trapattoni paved the way for disappointment when he told a Thursday news conference it would be impossible to replay the match.
"I cannot ask this because I know it is impossible. When a referee decides, it ends the game for me," he said.
The Italian instead criticised FIFA's late decision to seed the playoff draw and called on the ruling body to scrap extra-time in future playoffs as it handed an advantage to the team drawn at home in the second leg.
