The move sees PSA regain its position as Europe's second-largest automobile manufacturer, after Germany's Volkswagen.
The move sees PSA regain its position as Europe's second-largest automobile manufacturer, after Germany's Volkswagen, overtaking rival French firm Renault.
PSA said in a statement it was also buying GM Europe's financial operations for 900 million euros in a joint deal with bank BNP Paribas, taking the total value of the deal to 2.2 billion euros.
The takeover includes six assembly plants and five component-making facilities and some 40,000 employees.
Plans for the takeover of the Opel division by PSA, which owns the Peugeot and Citroen brands, were unveiled in the middle of February, sparking fears in Germany and Britain that the prospective new owner could cut non-French jobs.
The French giant's shares rose more than three percent on the Paris stock exchange early Monday on news of the deal, trading at 19.67 euros.
PSA boss Carlos Tavares said the firm was "deeply committed to continuing to develop this great company and accelerating its turnaround".
"We are confident that the Opel/Vauxhall turnaround will significantly accelerate with our support, while respecting the commitments made by GM to the Opel/Vauxhall employees," Tavares said.
Vauxhall employs around 5,000 people in Britain. Opel operates some 10 factories in Europe spread across six countries, and had 35,600 employees at the end of 2015, 18,250 of them in Germany.
Founded in 1862, Opel, with its lightning-bolt emblem, is a familiar sight on European roads, but in recent years the firm has booked repeated losses, costing Detroit-based GM around $15 billion since 2000.
Britain, where it sells vehicles under the Vauxhall brand, is Opel's largest European market.
A sharp fall in the pound since Britain's vote to quit the EU last June sank Opel's hopes of getting back into the black in 2016, and it ended up reporting a loss of $257 million.
Britain's Unite trade union said the productivity of the UK plants and the strength of the Vauxhall brand meant that it "makes sense" for PSA to continue manufacturing there.
Unite boss Len McCluskey also called on the British government to end the uncertainty surrounding trade relations with the EU after Brexit.
"We need every assistance from the government to give this sector a fighting chance," he said.
"That absolutely includes committing now to securing access to the single market and customs union."
PSA said the deal will enable substantial economies of scale and savings in purchasing, manufacturing and research, and they aim to return Opel-Vauxhall to profit in the next three years.
GM's chairman and chief executive Mary Barra said at a press conference on Monday that the sale had been "a difficult decision for General Motors ... but the right one".
In the statement confirming the sale, Barra hailed the move as "another major step" in the company's efforts to improve its performance.
"We believe this new chapter puts Opel and Vauxhall in an even stronger position for the long term and we look forward to our participation in the future success and strong value-creation potential of PSA through our economic interest and continued collaboration on current and exciting new projects," Barra said.
PSA said that all of Opel-Vauxhall's pensions would remain with GM, apart from a German pension pot and some smaller plans, which will be transferred to the French manufacturer.
GM is to pay PSA 3.0 billion euros for settlement of these obligations.