The fifth republic is the result of a coup by a military leader (De Gaulle) that was in talks with commanders of tank divisions to drive to Paris should he not be instated president and allowed to write his own constitution (written by Pierre Debré, a friend of his).
For those who can read French and who would want to learn more about this I heartily recommend the recently published Gouverner la France [1] (Governing France), a collection of books written by Michel Winock as part of the prestigious Quarto Gallimard series.
It includes titles like L'Agonie de la IVème République (Agony of the Fourth Republic), La fièvre hexagonale : les grandes crises politiques de 1871 à 1968 (Hexagon Fever: Major Political Crises from 1871 to 1968) and a pretty good biography of de Gaulle. It's from that book that I learned of all the craziness of 1958, the one that involved general Salan (who would be sentenced to death a few years after that for trying a coup against de Gaulle) and all.
Calling it a coup is highly debatable and debated. It happened outside of the scope of what the 4th happened, and it happened because the 4th had broken down and was not working anymore.
It should be noted that some of the main people who called to view it as a coup were people from other parts of the political spectrum who had other ideas of how it should happen and who should end up in power, including Mitterand (who ended up president of the 5th in 1986). These people however were also for the end of the 4th.
The military guys from Algeria had already put their hands on Corsica, plus, the iminent threat of the landing on the shores of Southern France (or via an aerial operation, can't remember exactly) of said military forces was heavily used by de Gaulle during the negotiations that got him into power.
It was ripe to burn since 1958.