The original deal envisaged 3.2 Telekom shares and $30 cash being exchanged for each Voicestream share - a mix that would have required a cash component of some $8bn.
Completion of the deal comes nearly a year after it was first announced. Telekom and Voicestream cleared their final regulatory hurdle earlier this month.
The deal, which was valued at $50bn (£35bn) when it was first announced, is now worth closer to $26bn.
Beleaguered Telekom chief executive, Ron Sommer, has staked his reputation on the success of the deal.
Earlier this week at the company's AGM, he faced shareholders made angry by a 70% fall in the company's share price during the past year.
Re-jigging the deal
Since the Voicestream deal was announced, Deutsche Telekom has been forced to re-jig the terms of the acquisition.
The original deal envisaged 3.2 Telekom shares and $30 cash being exchanged for each Voicestream share - a mix that would have required a cash component of some $8bn.
But in order to avoid being taxed in the US, cash paid to Voicestream shareholders could not amount to more than 17-18% of the total.
Therefore the mix has shifted toward more shares to keep the cash component within the tax-free threshold.
To finance the merger, Deutsche Telekom will issue 1.17 billion new shares, increasing its share capital by 2.99bn euros.
The new shares will also help Deutsche Telekom complete its purchase of Powertel.
$4bn cash payment
In addition, Telekom will make a cash payment of $4.23bn (4.97bn euros) to Voicestream shareholders.
The precise structuring of the deal has yet to be made public but Deutsche Telekom was due to announce details later on Thursday.
Analysts had previously speculated that falls in the German company's share price over the past year would lead Voicestream shareholders to demand a renegotiation of the deal - as, under certain circumstances, the original deal had permitted.
Expanding
Deutsche Telekom has said the acquisition of the US mobile phone operator will enable it to expand in existing and new markets.
It will also speed up development of new mobile phone products and services, the company said.
Many analysts have been sceptical about the deal, with some even saying that Deutsche Telekom should have ditched Voicestream.
Critics have argued that Telekom would have been better off building a more commanding presence in Europe than attempting to break into the US market, where technology standards are often different.
Because of the negative reaction, Mr Sommer's fate has been closely identified with the success of Voicestream.
He has also been under pressure from shareholders because of a 70% slump in Telekom's share price in the past year.
Earlier this week Mr Sommer faced thousands of angry shareholders at the company's AGM.
Shares rise
Shares in Deutsche Telekom jumped 5% on news of the deal's close, before slipping back slightly.
At 1140 GMT, the shares were nearly 4% higher at 25 euros.
Analysts said they were pleased to see the company tilt the financing of its Voicestream acquisition towards equity, relieving pressure on its debt pile of 60bn euros.
Also on Thursday, Deutsche Telekom closed its acquisition of Powertel and confirmed that it would be selling its stake in Sprint FON.