The bout for the IBF welterweight title has already been cancelled twice due to both men suffering injuries in training.
Brook's hopes of setting a third date then seemed to be in doubt when pound-for-pound king Floyd Mayweather announced Alexander as his next preferred fight.
However, with Mayweather settling on Robert Guerrero as his next opponent, that has cleared the way for unbeaten Sheffield fighter Brook to finally get in the ring with Alexander.
"It's been a frustrating wait for the fight to be on, but now I'm ready to get in there on May 18 and bring that title home," said Brook, who is the mandatory challenger to Alexander's crown.
"We've both had to pull out through injury so we are both hungry for the fight, but I believe I will prove I've got a bigger appetite for the battle and that I'll be too good for him."
The initial date for the fight on January 19 was scuppered when Brook suffered an ankle injury in December last year.
Alexander was then forced to pull out of the February 23 date due to a bicep complaint.
Brook puts his perfect 29-0 (19 KOs) record on the line when he goes to America for his maiden world title tilt.
Some bookmakers see the fight as even but many experts predict the younger but more established American, who has also won a world title in the light-welterweight division, will have too much for the British challenger.