Christian Bale played the serial-killing, style-obsessed character in a 2000 movie.
The Hayes had sold about 150 tickets when what was described as a "licensing glitch" became apparent last November.
Producer Bradley Barrack's company, BB-Arts, had a verbal agreement to licence the show with its writers – Roberto Aguirre-Sacasa, who wrote the book, and Tony Award winner Duncan Sheik, who wrote the music and lyrics.
But when publisher Samuel French took over as licensee, that agreement was superseded.
Both Barrack and Hayes' chairwoman Lisa Campbell were confident the issue would be resolved quickly.
But Barrack, who is also assistant company manager on The Book Of Mormon, has been unable to lock down the rights in time.
"I'm very disappointed," he said. "We were trying to get in early with it before anyone else had done it. I think I jumped the gun trying to get in there too quickly."
Barrack hopes further negotiations will allow the musical to run at the Hayes next year.
"It was going to be a pretty unique style of show for Australia to see," he said. "It's a very dark piece."
While actors had been discussed, the musical had yet to be cast when ticket sales were suspended.
The Hayes' general manager, Victoria Wildie, has advised ticket buyers they can either transfer to Cry-Baby with a free program or get a refund.
"We are very sorry that this turn of events has come about and for the time that it has taken to reach this decision," she wrote. "We have been holding out hope that the licensing issues would be resolved within a timeframe that would allow the production of American Psycho to go ahead as planned but it was felt that the time had come to make a decision to withdraw rather than keep the season in limbo any longer."
Cry-Baby will have the same creative team as American Psycho, headed by director Alex Berlage. Centring on a teenage rebel in 1950s Baltimore, it starts its run on July 20.
Garry Maddox is a Senior Writer for The Sydney Morning Herald.
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