Roark prevailed over rival bidder TriArtisan Capital Partners in the later stage of the process, according to people familiar with the matter.
The deal values CKE at between $1.65 billion to $1.75 billion, the people said.
Apollo declined to comment on the transaction. Goldman Sachs Group Inc and Wells Fargo & Co represented CKE in the deal.
TriArtisan was in the lead to buy CKE in an auction that also drew interest from Roark Capital and Pamplona Capital Management, but discussions were continuing and Apollo could still turn to another buyer, Reuters had reported earlier.
The company, with 3,400 restaurants in 42 U.S. states and 29 countries, competes with fast-food companies such as McDonald’s Corp and Burger King Worldwide Inc.
Atlanta-based Roark has invested in more than a dozen restaurant concepts, including Miller’s Ale House, Arby’s, Auntie Anne’s, Cinnabon, Carvel Ice Cream, Corner Bakery, Il Fornaio, McAlister’s Deli, Moe’s Southwest Grill, Schlotzsky’s and Wingstop.
Olivia Oran is a reporter for Reuters News in New York