Cardiff agreed a fee with FC Copenhagen last week, believed to be in the region of £7.5m, for the 20-year-old striker.
Cornelius will join up with his team-mates at the Premier League newcomers next week after a short holiday.
"This is a dream come true for me," Cornelius told the club's official website: "I've been given a great opportunity and this is a very exciting project that's taking place at Cardiff City.
"I'm very happy to be here and to be playing for this club in the Premier League.
"There were opportunities for me this summer and I had to think about what was the right thing to do, the correct step for me to take in terms of my career.
"I was impressed by what the club is trying to do, the fact it has reached the Premier League and that it has a passionate fan base supporting it."