The 26-year-old Sheffield fighter will take on the American in Las Vegas on January 19 after promoter Eddie Hearn successfully negotiated with Alexander's team.
Brook (29-0, 19 KOs) has gradually manoeuvred himself into the mandatory challenger position with a series of solid wins over the last 18 months and gets his reward when he takes on decent but unspectacular champion Alexander, 25, in the United States.
The Yorkshireman said: "We're finally there. This is what I've dreamed of since I was nine years old - fighting for the world title.
"I'm so passionate and hungry for this world title. Nothing's going to stop me. I can't wait to get in that ring, in America, and get that world title.
"Fighting for the belt in America is going to make it a lot more special. I could have had it my own way, in Sheffield for example, but to be honest I don't think he would have come over here.
"So I'm going to go over there, to his country, and capture a world title. Imagine that, when I'm stood in that ring, being crowned world champion after beating him in his own back yard after I've dreamed of it for all these years, hearing the words 'and the new.....!"'
Brook added: "I don't mind it being in America. I've agreed to fight him in his own country and whether it was Vegas or New York, it's all going to be nice fun and games for me when I get in there and do what I'm going to do - take him out."
St Louis native Alexander, whose record reads 24-1 with 13 early wins, beat Randall Bailey in October to win the IBF crown.
It is understood the MGM Grand Hotel and Casino is the likeliest venue for the bout.