Outerwall shares rose 11 percent to $52.10 in premarket trading on Monday, slightly above the cash offer price of $52 per share.
The deal comes about four months after Outerwall, under pressure from activist investor Engaged Capital, said it would explore strategic and financial alternatives.
The hedge fund urged the company in February to consider strategic options, including going private.
Engaged Capital, Outerwall's second-largest shareholder, had a 14.1 percent stake in the company as of June 2.
Outerwall, earlier known as Coinstar, gets most of its revenue from the Redbox business it acquired in 2008. The company also has a phone recycling business, called ecoATM, that Engaged Capital had urged it to sell or shut down.
Morgan Stanley is Outerwall's financial adviser for the deal. LionTree Advisors, Bank of America Merrill Lynch, Barclays, Credit Suisse and Jefferies are advising Apollo.
Bank of America/Merrill Lynch, Jefferies Finance, Barclays and Credit Suisse are providing the financing for the deal.
Up to Friday's close, Outerwall shares had risen about 28 percent this year.