The duo have signed long-term deals at St James' Park, but will spend the season on loan at Forest.
Central defender Lascelles, 20, told the club's official website: "As soon as I found out about the move I was over the moon.
"It's a huge step up for me. Newcastle are a club with a massive fanbase, a big history and an unbelievable stadium.
"First, I am going back to Forest for another season and that will be great for my development. Hopefully I can come here next summer an even better player."
Goalkeeper Darlow said: ""I'm delighted for it all to be sorted and I can't wait to get started here at Newcastle.
"The loan back is going to be another year, but it gives me time to develop my game further before I return. "I'm looking forward to the future."
"It is a real coup to have signed two of the finest young talents in English football outside of the Premier League," said Newcastle manager Alan Pardew.
"Jamaal and Karl will be a big part of the future of this Club. They have both performed outstandingly in the Championship over the last couple of years and I have no doubt they will be able step up to this level once they return from their loan at Nottingham Forest."
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- Nottingham Forest