To quote (loosely) from my own book on the topic, as far as we can tell, it happened in the late third century BCE, when the original Hebrew script was passed over in favor of the Aramaic script by the Sanhedrin (rabbinical assembly) as its script of choice for the biblical canon then being compiled and transcribed.
To downplay the historic significance of what they were doing, its name was changed, too, from ktav ashuri (“Assyrian script”) to the more neutral ktav merubá (“Square script”)—by which name it is still known today.
It was an extraordinary decision, nonetheless. It was to be expected that the Aramaic language and script be used in the civil administration of the new Persian province of Judea—but to use that same script for the biblical canon and the Talmud instead of the one used by Moses, David, and Solomon is astonishing. Not for nothing is it one of the best kept secrets in history.
Today such a move would be unthinkable, but we should remember the historical context of the Sanhedrin’s decision: as the official imperial language and the lingua franca of most of the Middle East for nearly five hundred years, Aramaic was the Latin of its time in this part of the world – with all the prestige and significance that this implies. Furthermore, it could be argued that adopting the Aramaic script was not a complete renunciation of the Old Hebrew: although the forms of its letters had changed considerably from their ancestral Canaanite origins, in name and function they were still the same characters. In addition, since no great body of work had been committed to or at least had survived in the old script, there was nothing much to lose in terms of a written cultural heritage. Finally, since Old Hebrew was essentially the same script used by the Phoenicians and all other Canaanites, it could be argued that there was nothing uniquely Judean about it anyway.
To all this I would add yet another argument, for which there is no explicit evidence in the sources, but it is compelling nonetheless: the Aramaic script was graphically simply more practical and better suited to the task of large-scale documentation. With its clear discipline of forms designed around a square template, it was certainly developed with this in mind by generations of Mesopotamian scribes. By contrast, the Old Hebrew alphabet, which had been typically limited to comparatively short texts on stone or clay, had little “rhythm” or consistency, much less compact, and suffered from poor or ill-defined distinctions between certain letters:
Nevertheless, the topic was clearly a sensitive one, and discussions on the subject kept to a minimum. Early Talmudic debates steered a wide berth around it altogether.
To settle the matter, it was decided to adopt the suggestion of Rabbi Ḥisda—namely, that the Old Hebrew script is in fact “ktav libonaah”. The meaning of libonaah is a mystery: it isn’t Hebrew—its spelling doesn’t even comply with Hebrew grammar—but it sounds distinctly pejorative (the Hebrew word honaah, for example, means “deception”), and it is possible they deliberately used a Babylonian code word that future generations wouldn’t understand. At any rate, the way it was interpreted in the final resolution was that it belonged to the common, i.e., non-Jewish, inhabitants of the land:
Originally the Torah was given to Israel in the Hebrew script and in the sacred language [Hebrew]; in the time of Ezra the Torah was given in the Assyrian [Aramaic] script and the Aramaic language. They chose for Israel the Assyrian script and the Hebrew language, leaving the Hebrew script and the Aramaic language for the ordinary people.
The implicit message was therefore clear, not to say intimidating: “Yes, the Torah was given in the Hebrew script originally, but now it’s in the Aramaic script—deal with it, or consider yourself not one of us.”
And so it came to pass that the original Hebrew script was “kicked upstairs” to largely ceremonial roles, such as depicting the name of God in biblical scrolls, coin inscriptions invoking the heroic Israelite kingdoms of old (see b in illustration—courtesy of J. Naveh), and so on. In a revival of this practice, some of the coins of modern Israel also boast a word or two in the old script on the back (a in the illustration):
But today, even educated Israelis cannot read it, nor, oddly enough, are they at all curious as to what it says. The letters on the modern sheqel coin, for example, spell “YHD,” i.e., Yehud (the Persian imperial name for the province of Judea), yet most Israelis have no idea what it says; if pressed, they assume it says sheqel.
None of this is taught, or even hinted at, in modern Israeli education. Like my friends and everyone else I know, I too assumed, in my teens, that the Old Hebrew alphabet was simply an earlier incarnation of the familiar forms of Square Hebrew. Only occasionally did I puzzle at the absence of any intermediate stages to explain the dramatic differences in some cases, or indeed why older texts were sometimes more recognizable than texts from later periods. It is a measure both of the historic sensitivity of the subject and of the recent greater security in the nation’s cultural identity that in the past few years the Old Hebrew script has been tentatively introduced into the Israeli school curriculum, albeit still in a very limited and somewhat gimmicky fashion (“At the camp, the children will learn to write their names in ancient Hebrew and come away with a little scroll,” to quote a schoolteacher in a radio interview in 1995). Even then, however, nary a whisper is made of its unceremonious dumping in favor of a younger and once glamorous sister script.
(Originally written in reply to a question at Quora.com)