The $130bn (£84bn) deal was announced by Vodafone after the close of trading on the London Stock Exchange.
The company will return £54bn to its shareholders, of which £22bn 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 chief executive Vittorio Colao told BBC News: “We got an offer that we thought was in the interests of our shareholders to accept – at the end of the day it’s as simple as that.”
The company is launching a £6bn 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.
Analysts said that the funds would also allow Vodafone to invest in Europe.
“I think Vodafone has looked at its European market and decided it needs to consolidate its business in a number of countries by buying up broadband and cable TV assets to go with its mobile businesses,” said Mark Newman, telecoms analyst at consultancy Informa.
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 $5bn in tax in the United States.
Not paying any tax on the deal may be controversial.
“Everybody is entitled to his or her opinion… but we don’t deal with opinions, we deal with rules, and with standard rules and practices and tax practices from any jurisdiction where we operate,” Mr Colao said.
“This jurisdiction is the Dutch one but even the UK one has very similar rules – we apply the rules.”
Although the deal will not give the UK a direct tax windfall, it could provide a boost for the UK economy in a number of ways.
Vodafone will invest some of the money in developing its business, of which a significant part is in the UK.
Shareholders will receive a payout, which they might choose to spend or invest.
They will also pay tax on the dividends they receive.
BBC Business Editor Robert Peston said: “Possibly it would be wrong to carp and wring hands that Vodafone won’t be paying a penny of tax to the British taxman on the tens of billions of pounds of profit it will make from the disposal.
“Because if it had been obliged to pay very substantial tax on the sale, it would have turned down the offer from Verizon Communications – and a windfall for the British economy would have been lost.”
Nevertheless, the tax affairs of big companies have been under scrutiny, particularly since it emerged that Google, Starbucks and Amazon had found legal ways to pay relatively little tax on their big operations in the UK.
Margaret Hodge, who questioned executives from those firms as chair of the Public Accounts Committee, told the BBC: “I think we want reassurance that HMRC is doing its darndest to look over this deal in huge detail to make sure that Vodafone, under the existing law, is paying all tax due.”
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.
Vodafone shares rose 3.4% during the day.