The move will let PSA regain its position as Europe's second-largest automobile manufacturer, after Germany's Volkswagen (IOB: 0P6N.IL - news) , overtaking its rival French firm Renault (LSE: 0NQF.L - news) .
PSA said in a statement it was also buying GM Europe's financial operations for 900 million euros in a joint deal with the bank BNP Paribas (LSE: 0HB5.L - news) , taking the total value of the deal to 2.2 billion euros.
The takeover includes six assembly plants and five component-making facilities, and about 40,000 employees.
Plans for the takeover of the Opel division by PSA, which owns the Peugeot (Other OTC: PUGOF - news) 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 carmaker's shares rose 2.7 percent on the Paris stock exchange on Monday to 19.58 euros. Shares (Berlin: DI6.BE - news) in GM were down around 2.1 percent to $37.43 in midday trading in New York.
PSA chief executive Carlos Tavares said the company 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 across six European countries, and had 35,600 employees at the end of 2015, of which 18,250 were in Germany.
Founded in 1862, Opel, with its lightning-bolt emblem, is a familiar sight on European roads, but in recent years the company 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 profitability in 2016, and it ended up reporting a loss of $257 million.
- 'Difficult decision' -
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 would enable substantial economies of scale and savings in purchasing, manufacturing and research, and the company aims 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 a 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.
GM said that the sale would force it to account for deferred tax assets and pension losses and that it would make a one-time charge of up to $4.5 billion.