NEW YORK — Sprint, the No. 3 cellphone company in the U.S., is selling a 70 percent controlling stake to Japan’s Softbank for $20.1 billion.

The deal, announced Monday 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.