Krispy Kreme Doughnuts Inc. will be acquired for about $1.35 billion by JAB Holding Co., the European investment fund with a fast-expanding stable of famous brands ranging from Caribou Coffee to Jimmy Choo shoes to Durex condoms.

The Winston-Salem, N.C.-maker of hot, glazed doughnuts would be taken private in the deal.

JAB, the money manager...