Does the Pope teach infallibly ONLY when he speaks ex cathedra? The Vatican Council (opened on 8 December, 1869) defined as "a divinely revealed dogma" that "the Roman Pontiff, when he speaks ex cathedra — that is, when in the exercise of his office as pastor and teacher of all Christians he defines, by virtue of his supreme Apostolic authority, a doctrine of faith or morals to be held by the whole Church — is, by reason of the Divine assistance promised to him in blessed Peter, possessed of that infallibility with which the Divine Redeemer wished His Church to be endowed in defining doctrines of faith and morals; and consequently that such definitions of the Roman Pontiff are irreformable of their own nature (ex sese) and not by reason of the Church's consent"1 - Source: Explanation of papal infallibility | Infallibility | New Advent.

1. cf. Denzinger 1839 And so We, adhering faithfully to the tradition received from the beginning of the Christian faith, to the glory of God, our Savior, the elevation of the Catholic religion and the salvation of Christian peoples, with the approbation of the sacred Council, teach and explain that the dogma has been divinely revealed: that the Roman Pontiff, when he speaks ex cathedra, that is, when carrying out the duty of the pastor and teacher of all Christians by virtue of his supreme apostolic authority he defines a doctrine of faith or morals to be held by the universal Church, through the divine assistance promised him in blessed Peter, operates with that infallibility with which the divine Redeemer wished that His church be instructed in defining doctrine on faith and morals; and so such definitions of the Roman Pontiff from himself, but not from the consensus of the Church, are unalterable.

The question is, does the Pope teach infallibly ONLY when he speaks ex cathedra? Are there instances when a Pope has taught infallibly but not precisely as according to the definition by the Vatican Council? If so, what are some examples?
Ex cathedra is the means by which a pope defines dogma.
Infallible means "not prone to err."

The First Vatican Council says a pope cannot err when defining dogma. This does not imply he's prone to err when not defining dogma, nor does it mean he must speak ex cathedra to be inerrant.

Encyclicals, for example, are inerrant; otherwise, the Church would demand the faithful to assent to what might contain error, and the Church cannot lead one into error because the Church is indefectible. From Pope Pius XII's Humani Generis:


  
  Nor must it be thought that what is expounded in Encyclical Letters does not of itself demand consent, since in writing such Letters the Popes do not exercise the supreme power of their Teaching Authority. For these matters are taught with the ordinary teaching authority, of which it is true to say: "He who heareth you, heareth me";[Luke, X, 16] and generally what is expounded and inculcated in Encyclical Letters already for other reasons appertains to Catholic doctrine. But if the Supreme Pontiffs in their official documents purposely pass judgment on a matter up to that time under dispute, it is obvious that that matter, according to the mind and will of the Pontiffs, cannot be any longer considered a question open to discussion among theologians.