The "back to back" agreement, which was confirmed on Thursday morning, involves the Scottish Government buying the Clydebridge and Dalzell plants from Tata Steel, and immediately selling them on to Liberty.
The sale will be done under the same terms with no cost to the taxpayer.
Liberty will take responsibility for reopening, operating and investing in the two sites as part of its wider strategy to build an integrated and sustainable steel business across the UK.
Business minister Fergus Ewing said: "When Tata Steel mothballed the Dalzell and Clydebridge plants, I said we would leave no stone unturned in the quest to find an alternative buyer.
"That is why we established a Scottish steel taskforce and why I am delighted that our support for the steel industry has paid off."
The taskforce was established to secure jobs and the future of steel manufacturing in Scotland after Tata announced 1,200 UK job losses in October.
Its main focus was on finding a new buyer for the two plants.
The government recently announced a cut in business rates at the plants from April for any new operator, provided they continue to use the sites for steel production.
Ewing added: "Liberty has a proven track record in the UK steel industry and has ambitions to make these plants viable and successful.
"It's a huge tribute to the calibre of the highly-skilled workforce, and the efforts of everyone on the taskforce, that Liberty recognise the tremendous potential of Scottish steel."
Sanjeev Gupta, executive chair of Liberty House Group, said: "This agreement saves two great facilities in Scotland.
"Now we must turn our attention to restoring these businesses to their former glory, steadily rebuilding their skilled workforces and customer base.
"Clydebridge and Dalzell will fit well into our vision for an integrated, flexible and sustainable steel sector, from recycled local scrap using renewable energy-making green steel to value added downstream and engineered products.
"We're grateful for the support of the Scottish Government and Scottish Enterprise in enabling us to reach this very positive historic agreement."
Bimlendra Jha, executive chairman of Tata Steel's Long Products Europe business, said: "We welcome this deal which opens the possibility of a resumption of steel processing in Scotland.
"This has been achieved with the determination and support of employees, trade unions and the Scottish Government all working together."
Grahame Smith, Scottish Trades Union Congress general secretary said: "This is excellent news and all partners involved in the Steel taskforce should be congratulated for their hard work and tenacity.
"The workforce wasn't prepared to sit back and let the Scottish steel industry fade away and it is tremendously encouraging that the Scottish Government and its agencies were able to provide the support to secure a positive outcome."
Scotland's First Minister Nicola Sturgeon said: "I'm absolutely delighted that this deal has been sealed today - bringing to an end months of anguish and uncertainty for communities in Motherwell and Cambuslang.
"The concerted efforts of the Scottish Government, trade unions, the workforce and the companies themselves have paid off - and I would like to sincerely thank all of those involved for their patience and persistence over the last few months."