A £2.7million shortfall resulted in the Canadian city losing its right to stage the championships last month.
But a decision to allow Montreal to stage the championships was taken after an emergency meeting of FINA, the sport’s world governing body.
The event will now take place from July 17 to 31 as originally planned.
Athens, Berlin and Moscow had been in the running to take over the hosting of the championships.
Montreal was set to become the first city to lose a major championship since London was dropped as host of the 2005 World Track and Field Championships in 2001.
