At What Point in History Did Popes First Become Infallible?

 Popes officially became infallible in 1870 when the First Vatican Council defined the doctrine of papal infallibility, which states that the pope is infallible when he speaks ex cathedra (from the chair) on matters of faith or morals. This formal definition cemented an idea that had been part of church history and debate for centuries. 

• Formal definition: 

  The First Vatican Council (1869–1870) adopted the decree Pastor Aeternus, which established the dogma of papal infallibility when the pope is speaking in his official capacity to define a doctrine on faith or morals for the entire Church. 
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    The first 'infallible Pope' was Pius 1X in 1870.