The deal sees Macquarie take over the management and supervision of approximately £4 billion (€4.38 billion) worth of green infrastructure projects, including offshore wind farms.
The GIB has invested in offshore wind projects in the UK such as E.on’s 400MW Rampion site, the 630MW London Array site and Innogy’s 576MW Gwynt y Mor.
It has a target of investing £3 billion (€3.3 billion) in green energy projects over the next three years.
As part of the deal, the bank’s name will change to the Green Investment Group (GIG), and it will be able to make overseas investments.
Five independent trustees will have the power to approve or reject any proposed changes to GIG’s purposes in the future.
The consortium buying the fund comprises Macquarie Group, Macquarie European Infrastructure Fund 5 and the Universities Superannuation Scheme, a private pension pot for higher education institutions.
Macquarie’s head of investment banking Edward Northam replaces Shaun Kingsbury as group CEO, the investment firm announced alongside the deal.
"This new chapter provides the best of both worlds: a deep sector specialism coupled with access to a global platform and deep pools of capital," said Northam.
"We have ambitious plans for the growth of the Green Investment Group, starting with a continuation of our role as a leading investor in the UK and building on that through an additional international focus."
The UK government announced its plans to privatise the bank in June 2015. The then-UK-chancellor, George Osborne, said the sale would "enable it to access larger pools of capital".
Announcing the sale’s completion, the government today said privatisation would allow the bank to "increase investment in our green infrastructure as we transition to a green economy".
The government will continue to hold a stake in a small portfolio of GIB’s existing green infrastructure investments. GIG will continue to manage these assets until they can be sold on, the government said.
Claire Perry, the UK's climate change and industry minister, said: "We led the world in setting up the Green Investment Bank and it is now being copied by others.
"Now that it is in the private sector, it will be able to operate on an international level to tackle the global challenge of climate change.
"It is also perfectly placed to help us finance green initiatives for our Clean Growth Plan and realise the commitments set out in the Paris Agreement," she said.