The books of the Bible were already authorized by the first century.
There was no "Bible" in the 1st century. The closest thing to a Bible in the first century would have been the Septuagint; something Christians used extensively as most of the New Testament quotations of the Old are often taken verbatim from the Septuagint.
However Jewish opinion in the 1st century was quite diverse, perhaps you are familiar with the disagreements between the Sadducees and the Pharisees; the Sadducees accepting only the five books of the Torah; while the Pharisees accepted the Torah, the Prophets, and the Psalms.
With the fall of the Temple in the year 70 Sadduceeism died off, it was Pharisaism that survived. But there was no clear boundaries for the Jewish Canon, debates in Judaism over books like Daniel were going on independently of debates in Christianity over the nature of what is and isn't canonical.
The Torah and Prophets were probably pretty well established in the Pharisaical tradition, as was the position of the Psalms--what remained was that category now known as the Writings, and as such the basis of the Jewish Tanakh and its three-fold division of Torah, Prophets, and Writings. In 1st and later century Judaism the book of Daniel, as an example, wasn't among the Prophets, but among the Writings.
Christians certainly did receive their scriptural tradition from Judaism, but the boundaries are imprecise. For example among those texts that in Judaism are considered "the Writings" Christian writers regularly refer and quote from the Psalms, which were indeed well established as scripture in the period, but simultaneously we find some texts simply are never quoted or referred to, books whose status were indeed nebulous, such as Esther or the Song of Songs.
In Judaism the Greek textual tradition of the Septuagint was ultimately rejected, and a leading defining factor was the text having a known Hebrew original; which in antiquity meant Judaism excluding works such as Sirach and 1 & 2 Maccabees. Though it is now believed that many of these works did have a Hebrew or Aramaic original prior to being translated into Greek vis-a-vis the Septuagint.
So, as it stands, no there was nothing emphatically established by the end of the first century; there was instead a rough idea, a nucleus of scripture that was read and received within the nascent communities of the Church. It is in this context that 2 Peter refers to the writings of Paul being read as scripture in the churches. This is the burgeoning of a distinctively Christian scriptural tradition running independently and in parallel with the still developing Jewish scriptural tradition.
It is in that context which a canonical nucleus comes about; but remains a continued point of disagreement among Christians for hundred years. The conversation was chiefly friendly in antiquity, Christian leaders expressing that some accept X and while others accept Y, that everyone accepts Z, and so on. This can be seen throughout the writings produced throughout antiquity and into the medieval period. Local councils which did engage in which books were to be read in churches in their diocese such as the Councils of Laodicea and Carthage were not in complete agreement with one another--and we shouldn't expect they had to be given that they were local synods dealing with local issues of church practice.
The point remains that the Bible, as we understand it, is a liturgical document arising out of an evolving consensus. There was no prophet, no apostle, no angel, or anything of that nature which placed a stamp or seal on a collection of scriptures as the definitive biblical canon. It was instead a complex, organic process that could often be quite messy. There is no divinely ordained Canon, the Canon is a product of the Church and part of the Church's tradition. The issue of faith is that these texts, received into and by the Church and placed into a biblical canon are divinely inspired--it is a position of faith that says yes, they are. There exists no divine record of what is and is not to be canonical, that process occurred within the day-in and day-out life of the Church over many hundreds of years. And it is why, to this day, there is no universal agreement as to what is and is not canonical Scripture--namely the disagreement over the Deuterocanonical texts: Tobit, Judith, 1 & 2 Maccabees (& 3 Maccabees), the 151st Psalm, Baruch, etc.
-CryptoLutheran
