Some interesting answers...
This depends on what you mean by "bible". Let me offer two definitions:
1) a collection of Scriptures, including at least all of the 66 books of the "minimal" canon, compiled under one cover and in one language
2) As above, except that only those Scriptures authorized (by agreement, not by an individual) for inclusion are to be included.
The earliest known bibles satisfying definition 1 are Codex Vaticanus and Codex Sinaiticus, Greek "bibles" from the mid-4th century C.E. Vaticanus (dated slightly earlier, but Sinaitucus could be older) omits 1 & 2 Maccabees. Sinaticus includes Scriptures in the New Testament not normally included in any modern bibles. The reason: each of these was probably compiled under the direction of an individual, who chose what Scriptures to include.
In 393 C.E .the synod of Hippo agreed on which Scriptures were authoritative, and in 397 C.E. this decision was approved by the much larger council of Carthage. Using these authorizations, the Latin Vulgate (first produced about 405 C.E. or shortly after) was the first bible following definition 2. Note that the term "bible" (meaning "the book") as applied to a religious text first arose from Latin, so in fact the Latin Vulgate is the first bible **called** the bible (meaning "the book"). It should be noted that the Greek term "bibles" (books) was applied to the individual books of both Christian and Hebrew Scriptures prior to this time, but the first book identified as "the bible" was the Latin Vulgate.
http://www.bible-researcher.com/carthage.html
Note: the book of Baruch is missing from the list above. To the best of my knowledge, Baruch was first *formally* added to the biblical canon - by name, that is - by the 39 Articles in 1563. It was also added formally in 1564 when the decisions of the council of Trent were approved.
http://www.bible-reviews.com/charts_scriptures_d.html#Books
This seems very late to me, and I am hoping someone can direct me to evidence that Baruch was formally included, by name, at an earlier date. It is interesting, however, that our oldest complete Latin Vulgate, Codex Amiatinus, omits Baruch but includes the Letter of Jeremiah. This Codex is from the 8th century, so it would seem that Baruch was not formally authorized until after that time.
As for a "minimal canon" bible (sometimes erroneously referred to as a "Protestant" bible), the first "complete" bible (in any language) produced without the Scriptures of the Apocrypha was done by the Puritans in the late 1590s (they used the Geneva Bible). Source: HarperCollins Bible Dictionary. This practice caught on with other Protestant sects and in other languages afterward.
Jim