France cancels the concordat
When the new pope made clumsy moves to re-assert his authority over France, "the eldest daughter of the Church", the French National Assembly replied by cancelling the concordat and returning to the Revolution's separation of church and state.
(The French president wasn't the only one dictated to by Pius X, as if he still ruled Rome. In 1910 the pontiff also tried to set conditions for granting an audience to the American president: that he cancel his planned visit to a Methodist church in Rome. Teddy Roosevelt was an ebullient self-made man, and Head of State of a place a trifle bigger than the Vatican.... He, too, refused. )
Incensed at what they considered interference in French affairs, the National Assembly recalled their ambassador to the Vatican.  Pius X responded by dismissing the bishops of Dijon and Laval who were noted for their republican sympathies. This was a violation of the Napoleonic concordat which permitted the French head of state to name the bishops, who were only then to be invested by the pope, (Articles 4 and 5 of the Concordat of 1801).
It was this attempt by the pope to assert power over the French Church which completed the rupture. On 30 July 1904 the French Government announced that it was breaking off diplomatic relations – and, with that, the concordat became a dead letter. 
For the republican majority, it was not a matter of getting rid of religion. Instead, they wanted to limit the power of the Catholic Church which was allied with the royalists, and broaden their support from other confessions, especially the Protestants. 
The French break in diplomatic ties to the Vatican was followed the next year by the official cancellation of the concordat. While the Protestants and Jews complied immediately with the 1905 Law of Separation, some Catholics condemned it, not on the grounds that it removed Church privileges, but because, according to them, it promoted "state atheism". After the concordat was cancelled by the law separating church and state, Pius X did his best to limit the damage.
The new pope feared that the separation would injure his prestige and serve as a 'bad example' elsewhere, especially in Spain. In his first encyclical, Vehementer nos, dated 11 February 1906, he condemns the very principle of separation, deplores the unilateral abolition of the concordat and, as he saw it, the way the Law called into question a fundamental principle of the Church which was 'essentially an unequal society, that is, a society comprising two categories of persons, the Pastors and the flock' . 
However, there may have been a bit more to this encyclical than a call to faith. Documents from the Vatican Archives are said to show that Cardinal Merry del Val was hoping that a "national uprising" of French Catholics would force the government to negotiate a new concordat. 
This ambition puts in a rather unpleasant light the pope's inflamatory ending to his encyclical. He feels the "keenest sorrow [to] observe that the French Government has just done a deed which...seems to be calculated to plunge the whole country into disorder".  His encyclical, he assures the French, is meant as "comfort to you in the midst of the terrible calamities through which you will have to pass."  And the pope exhorts his flock (meaning everything metaphorically, of course): Under the leadership of the "your priests, your bishops, and above all with this Apostolic See ... armed for the fray, go forth fearlessly for the defense of the Church". 
Luckily both the "keenest sorrow" that consumed him and the "comfort " that he so kindly offered proved totally superfluous. The cancellation of the concordat unleashed absolutely nothing. In a largely calm and peaceful manner this "eldest daughter of the Church" became a permanently secular state. 
1. Richard Cavendish, "Pius X Elected Pope" [4 August 1903], History Today, Volume: 53, 8 August 2003. p. 52. Teddy Roosevelt is said to have climbed trees while members of his cabinet stood below entreating him not to do so in front of the press. Not one to be impressed by titles.
2. Alain Gresh, "Aux Origines Des Controverses Sur La Laïcite", Le Monde Diplomatique, August 2003, Pages 18 -19. [The President of France was Émile Loubet, and the King of Italy King Victor Emmanuel III.]
5. Jean Paul Scot, "Comprendre la loi de 1905", Conférence de Jean Paul Scot, Versailles, 30 November 2005, Section II.3. http://www.histoire.ac-versailles.fr/old/histoire/1905/jpscot.htm
6. Gresch, ibid, quoting Pius X, "Vehementer nos", 11 February 1906, Paragraph 8.
7. Comprendre la loi de 1905, Conférence de Jean Paul Scot, Versailles, 30 November 2005, Section II.3. http://www.histoire.ac-versailles.fr/old/histoire/1905/jpscot.htm
8. Pius X, "Vehementer nos", 11 February 1906, Paragraph 12. http://www.newadvent.org/library/docs_pi10vn.htm
9. Pius X, ibid., Paragraph 16.
10. Pius X, ibid., Paragraph 17.
11. Jean Paul Scot, ibid.