The $23-a-share deal comes a week after the companies confirmed news reports of takeover discussions, jolting cinema-operator stocks.
His counterpart at Regal, Amy Miles, said: "We are excited to have reached an agreement with Cineworld, at a price that represents a meaningful premium on Regal's unaffected share price for our shareholders".
Cineworld will gain opportunities for growth outside Britain, where consumer spending is falling ahead of the country's exit from the European Union.
Commenting on the acquisition, Mooky Greindinger, chief executive officer of Cineworld, said:"Regal is a great business and provides Cineworld with the optimal platform on which we can continue our growth strategy".
Regal shares have risen 13.6 percent since Reuters first reported in November that Cineworld had approached Regal over a potential deal. The stock is down 19 percent since news of the talks emerged.
Denver billionaire Phil Anschutz is selling his movie theater chain. Regal Cinemas recently said it will experiment with charging more for tickets during peak times and less at times when attendance is lighter.
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Cineworld has been refurbishing movie theaters in the U.K., Europe and Israel to offer "top of the class" facilities, Greidinger said, including wider seats, a variety of food offerings, better sound systems and bigger screens.
Crucially, the deal gives Cineworld entry into the North American market.
Regal's theater brands include Regal Cinemas, United Artists and Hollywood Theaters.
The acquisition price of $23 per Regal share values the entire issued and to be issued share capital of Regal at $3.6 billion or about 2.7 billion pounds, with an implied enterprise value of $5.8 billion or 4.3 billion pounds. During the same period last year, the company posted $0.29 EPS. equities research analysts predict that Regal Entertainment Group will post 0.95 earnings per share for the current year. "The long-term investment proposition has fundamentally changed as a result of higher debt and earnings becoming heavily dominated by mature markets", the analysts said.
The deal will lead to Cineworld's expansion in the U.S. and will create the world's second largest cinema group with a total of 9,542 screens, 7,315 of these in the United States. He joined Cineworld when it bought his family theater chain in 2014.
But some Cineworld shareholders have expressed concern that the group will overstretch itself by expanding into the United States through the takeover. Regal unsuccessfully tried to sell itself in 2014.