The ending was dramatic and brutal. A right uppercut had Kalocsai reeling, and Schwer pursued him across the ring, connecting with two big lefts and finally a big overarm right, which left the challenger face down on the floor and sparked the referee's intervention without a count.
As the World Boxing Council's mandatory challenger Schwer is guaranteed a world title bout this year - as long as he keeps winning.
His manager, Mickey Duff, said: "He's due to get his chance in August, and if that fight comes off I have no doubt he will be the champion."
Kalocsai was the second replacement as Schwer's opponent but he had good credentials. He had taken Liverpool's Colin Dunne 12 rounds for the World Boxing Union title and had never been stopped or knocked down in his career.
But Schwer turned in a masterful performance, dropping his opponent with a short right in round five and shrugging off the effects of a small cut around his left eye in the fourth.
In his only world title challenge to date, Schwer was stopped on cuts by Rafael Ruelas in January 1995.
Schwer's next fight will be another step up in class - be it against the WBC champion, Cesar Bazan, or a mandatory European defence against the Spaniard Oscar Garcia Cano.
Schwer won the title by beating Cano in Zaragoza in 1997, rallying from a points deficit for a 10th-round stoppage. A rematch would be risky, but Schwer is utterly confident in his own ability these days.
Schwer said: "1999's going to be my year and it's all going to plan. I've been doing this long enough to know what it's all about. I am the Millennium Man."Reuse content