NEW YORK » Sprint, the No. 3 cellphone company in the U.S., is selling a controlling stake to Japan’s Softbank for $20.1 billion.

The deal, announced today in Tokyo, positions Sprint Nextel Corp. as a stronger competitor to U.S. market leaders Verizon Wireless and AT&T, but it doesn’t solve all of the company’s underlying problems.

Sprint, which is based in Overland Park, Kan. has been limping along since 2005, when it bought Nextel. The merger quickly turned sour, saddling Sprint with the cost of running two incompatible networks while customers fled.

Softbank Corp., a holding company with investments in Internet and telecom businesses, made its own venture into the wireless world in 2005, with the acquisition of Vodafone Japan. It turned that business around, giving President Masayoshi Son the confidence that he can make Sprint a profitable company again after five straight years of losses.

Sprint CEO Dan Hesse has laid the groundwork for a turnaround — the company’s reputation for customer service has improved during his tenure. But his efforts haven’t had an immediate impact on profitability. On its own, the company would have a hard road ahead, as it pays for both a network revamp and $15.5 billion in iPhones from Apple.

Under the deal, Sprint shareholders can turn in 55 percent of their shares to Softbank in exchange for $7.30 per share. Sprint shares were up just 3 cents at $5.76 in morning trading today, suggesting that investors had accurately pegged the value of the transaction last week, when they sent the stock up 14 percent based on reports of talks between Softbank and Sprint.

Softbank’s is paying $12.1 billion for the 55-percent stake. It’s buying an additional $8 billion worth of shares from the company, for a total stake of 70 percent. That investment will dilute the value of existing shares, and is the reason Sprint’s stock didn’t trade higher today.

“This is a transformative transaction for Sprint that creates immediate value for our stockholders, while providing an opportunity to participate in the future growth of a stronger, better capitalized Sprint going forward,” Hesse said.

Analysts were more reserved in their judgment.

“While we believe it will take far more than capital for Sprint Nextel to effectively compete with Verizon Wireless and AT&T Mobility, we believe the deal announced today, without question, strengthens Sprint’s position in the long-run,” said Christopher King at Stifel Nicolaus.

Kevin Smithen at Macquarie Capital said the deal doesn’t improve Sprint’s access to space on the airwaves, which is critical to improving its wireless data network, nor does it provide a path to improving its profitability. A merger with T-Mobile USA, the No. 4 carrier, might still be needed to deal with those problems, he said.

T-Mobile USA has its own plan: two weeks ago, it struck a deal to buy MetroPCS Communications Inc., the No. 5 carrier in the U.S.

Analysts also speculated that Softbank could buy out Clearwire Corp.’s shareholders and combine the company with Sprint. The Bellevue, Wash., company operates a wireless broadband network, but its network upgrades have been hamstrung by a lack of funds. Sprint already owns 48 percent of the company and resells access to its network as “Sprint 4G.” The speculation sent Clearwire’s stock up 11 percent today. It has doubled since Wednesday.

Softbank shares fell 5.3 percent on the news, as Japanese investors worried that the company is making a huge gamble. The shares have fallen by a third over a week. Standard & Poor’s had placed Softbank on “credit watch negative,” meaning its credit rating could be downgraded.

The deal has been approved by the boards of both companies. It still needs approval from Sprint shareholders and U.S. regulators. Softbank said the transaction is expected to be completed by the middle of next year.

Analysts say buying a foreign cellphone company makes little sense in terms of operational synergies. There’s little opportunity to improve service by combining networks or saving money by combining operations.

But Son said the U.S. and Japanese markets have much in common now that smartphones are all-important in both countries, and the two companies could benefit and learn from each other. By joining forces, Sprint and Softbank will become one of the world’s top smartphone carriers, gaining greater bargaining power with the manufacturers of the gadgets and network equipment suppliers.

Softbank was the first carrier to offer the iPhone in Japan. The iPhone has been such a hit in there that it has shaped Softbank’s brand image and helped it lure customers away from its two bigger rivals.

Son said Sprint and Softbank can work together to develop new technology for faster data networks. Softbank recently bought smaller Japanese rival eAccess, largely to gain access to the latest fourth-generation, or “4G” networks.

Son likes to take chances in a culture that doesn’t always reward risk. A graduate of the University of California, Berkeley, he was only 16 when he ventured alone to the U.S.

“I am happy to be able to tell you today of my big comeback to the U.S.,” he said. “This is going to be an even bigger challenge.”

Son has made no secret that he has been looking abroad for new growth as the Japanese mobile market has been stagnant for years. Softbank has been an exception in racking up strong profit despite such stagnation, largely due to the popularity of the iPhone.

The deal is the largest foreign acquisition ever by a Japanese company, and illustrates how the strong yen, which is usually seen as a negative for export-reliant Japan Inc., has boosted the overseas purchasing power of Japanese corporations as a stagnant domestic market pushes them to look abroad for growth.

Before today’s deal, the biggest overseas acquisition by a Japanese company was Japan Tobacco Inc.’s purchase of Gallaher Group of Great Britain in 2007 for about $19 billion.

The combination of Softbank and Sprint will tie with AT&T for world’s No. 3 mobile company by revenue after China Mobile and Verizon, according to Softbank.

The deal leaves three of the four national U.S. wireless companies with complete or substantial foreign ownership. Vodafone Group PLC of Britain owns 45 percent of No. 1 Verizon Wireless, and Deutsche Telekom AG of Germany owns T-Mobile USA outright.

Associated Press Business Writer Yuri Kageyama contributed from Tokyo.