The $130 billion deal was announced by Vodafone after the close of trading on the London Stock Exchange.
The company will return $84 billion to its shareholders, of which $34 billion will go to shareholders in the UK.
Vodafone will also invest money in its business, with funds earmarked for high speed mobile phone networks.
It said that by 2017 its main five European markets would have almost complete 4G coverage.
Vodafone group chairman Gerard Kleisterlee said: “The transaction will position Vodafone strongly to pursue our leadership strategy in mobile and unified communication services for consumers and enterprises, both in our developed markets and across our emerging markets businesses.”
The company is also launching a $9 billion investment plan called Project Spring, which will accelerate the introduction of 4G networks and increase investment in laying fibre optic cables, among other things.
The investments would allow the company to offer much faster broadband services to customers.
Project Spring will also add to Vodafone’s high street stores and develop mobile payment services.
Vodafone group chief executive Vittorio Colao said: “Project Spring will strengthen and accelerate our existing Vodafone 2015 strategy, enabling us to take even greater advantage of the growing global demand for ubiquitous high-speed data.”
It is the third biggest corporate transaction, behind Vodafone’s 1999 deal to buy Germany’s Mannesmann and AOL’s purchase of Time Warner in 2000.
Despite the huge size of the deal, it will not generate tax revenue for the UK.
Vodafone says that as the US business is owned by a Dutch holding company, it will not be liable for tax.
However, it will pay $5 billion in tax in the United States.
Vodafone shares rose 3.4% during the day.
The deal ends a long-running saga, with both Vodafone and Verizon trying to take full control of Verizon Wireless over the years, but having been unable to agree a price.
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