According to Bartleby, Berossus was a Babylonian priest-historian who wrote in Greek. As the Livius site states, Berossus' History of the Babylonians (the Babylonaica) is lost, but it was summarized by Alexander Polyhistor in C1 BC in a treatise of 42 books on world history and geography—which is also lost. That treatise, however, was used by Josephus (37–100 AD).
Josephus, 37–100 AD, discussed the gardens twice—once in Jewish Antiquities, and once in Contra Apionem (Against Apion, or Against the Greeks). The Project Gutenberg site offers the full text of Contra Apionem, and Jewish Antiquities. Or you can jump to the extracts pasted in brown below.
Josephus records that Berossus based his work on the archives in Babylon's Esagila temple, and that he repudiated the claims of Semiramis (accredited by Ctesias and Clitarchus), and claimed instead that Nebuchadnezzar built the gardens, in Babylon, for his wife Amytis. Be that as it may, it is commonly pointed out that the surviving Babylonian records contain no reference to the Hanging Gardens.
From Sacred Texts
19. ... When he [Nebuchadnezzar] had thus admirably fortified the city, and had magnificently adorned the gates, he added also a new palace to those in which his forefathers had dwelt, adjoining them, but exceeding them in height and splendor. Any attempt to describe it would be tedious: yet notwithstanding its prodigious size and magnificence it was finished within fifteen days. In this palace he erected very high walks, supported by stone pillars; and by planting what was called a pensile paradise, and replenishing it with all sorts of trees, he rendered the prospect an exact resemblance of a mountainous country. This he did to gratify his queen, because she had been brought up in Media, and was fond of a mountainous situation.
20. This is what Berosus relates concerning the forementioned
king, as he relates many other things about him also in the third
book of his Chaldean History [i.e. the Babylonaica]; wherein he complains of the Grecian
writers for supposing, without any foundation, that Babylon was
built by Semiramis, queen of Assyria, and for her false
pretense to those wonderful edifices thereto buildings at
Babylon, do no way contradict those ancient and relating, as if
they were her own workmanship; as indeed in these affairs the
Chaldean History cannot but be the most credible. Moreover, we
meet with a confirmation of what Berosus says in the archives of
the Phoenicians, concerning this king Nabuchodonosor, that he
conquered all Syria and Phoenicia ... (the rest of the text lists major events but no city or garden building)
—Joseph. contr. Apion. lib.1. c.19-20.
From Meta Religion
He [Nebuchadnezzar] also erected elevated places for walking, of stone, and made it resemble mountains, and built it so that it might be planted with all sorts of trees. He also erected what was called a pensile paradise, because his wife was desirous to have things like her own country, she having been bred up in the palaces of Media.
—Josephus, Antiquities of the Jews, bk X, ch 11, sctn 1