Hurd says the GIB has been a "real success story" since it was created in 2012 - becoming the world's first dedicated green investment bank and supporting nearly 100 green infrastructure projects in the United Kingdom and attracting £3 of third party funding for every £1 it invests.
The deal is expected to complete in two months and conditional on certain regulatory approvals including European Union merger clearance.
Caroline Lucas, co-leader of the Green party, said the safeguards secured by the government were "worthless" and accused the government of recklessness in its decision to offload GIB.
The British government set up GIB, which backs green projects with public funds, in 2012 as a commercial venture to spur private investment in green projects.
Climate Minister Nick Hurd this morning confirmed the deal after months of delays caused by heavy criticism from politicians and environmental campaigners.
Ministers have promised that the sale of the bank, which has invested in green projects from offshore windfarms to energy-saving street lights, will deliver value for taxpayers' money.
To protect the bank's green mission after privatisation, the government has established a "special share", held by a group of independent trustees, who will have the final say over any attempt to change the bank's core objective.
A number of GIB assets will be moved into the new offshore wind company which will be managed by it under via a 25% stake. "The UK will benefit from increased investment in our green infrastructure as we make the transition to a green economy".
The Green Investment Bank's chairman, Lord Smith of Kelvin, said in a separate statement from the bank: "Macquarie has made significant and important commitments to the United Kingdom government to maintain GIB as a discrete entity within its business. uphold GIB's green investment principles and report transparently on GIB's green impact".
Lord Smith of Kelvin, independent chair of the GIB said Macquarie's "significant and important commitments" showed it could be a good custodian of the bank, but warned it would be held accountable.
GIB will become the primary vehicle for Macquarie's renewable energy investment in the United Kingdom and Europe. Macquarie will utilise the market-leading expertise of the existing GIB team and will build on GIB's deep commitment to Edinburgh.
Macquarie Group's David Fass said the deal "will open further opportunities in low carbon investment both in the United Kingdom and further afield". It will also continue the collaboration between the bank and UK Climate Investments. He said: "We understand the responsibilities that come with this ownership, and are fully committed to maintaining its green objective as we grow the business".
Macquarie a year ago sold off Thames Water after investors reportedly banked profits in foreign accounts and dramatically cut the amount of taxes paid by the company.
"Under Macquarie's custodianship, the Green Investment Bank will operate in accordance with its green goal and green objectives and in line with the "special share" arrangements".