PSG came to be in 1970, according to its own logo, but it's actually a pretty complicated story which I am gonna try to tell.
When the amateur club of Saint-Germain (based in Saint-Germain-en-Laye, Yvelines) gained access to the 2nd Division at the end of the 1969-70 football season, the thinking heads at Paris F.C., a brand new club with zero infrastructures and no team, born from the desire of Parisians to (again) have a great club in Paris, thought it would be a good idea to merge with the amateur club in order to accelerate the climb to the French soccer elite (the 1st Division, now called "Ligue 1"). Well, it was a pretty good plan because, after a season in the 2nd Division, the Paris Saint-Germain F.C. gained access to the 1st Division for the 1971-72 season.
The problem is that the city of Paris didn't particularly like the name of the club and requested a change to Paris F.C. If the name change was not accepted, the club would have to say good-bye to its grounds (Parc des Princes, where PSG currently plays) and the monies. Long story short, the club refused the name change and ended up being split in two: Paris F.C. became the 1st Division professional team and Paris Saint-Germain was left with the amateurs and had to start again in the 3rd Division. That split happened in 1972, and that date could really be considered PSG's true birth year (if you ask me).
The irony is that Paris F.C. did go down the ranks of French football since the early 80s while Paris Saint-Germain has been in the elite of French football since 1974 (quite a feat). Paris F.C. is however slowly starting to raise from the depths and there might be hope for a second big club in Paris (again).