Here is the key excerpt from Josephus, courtesy of James Tabor's site:
Josephus on the Death of James brother of Jesus, in 62 C.E.
Book 20: chapter 9
CONCERNING ALBINUS UNDER WHOSE PROCURATORSHIP JAMES WAS SLAIN; AS ALSO WHAT EDIFICES WERE BUILT BY AGRIPPA.
Our interest here for the moment is the fact that James' death is not recorded in the Book of Acts, (just as the sack of Jerusalem and the destruction of the Temple is not recorded), and this suggests not only that Acts was written before 70 A.D., but actually before 62 A.D., since it is hardly conceivable that Luke would have left out the death of James or the destruction of the Jerusalem Church, had he written after that time.
"1. AND now Caesar, upon hearing the death of Festus, sent Albinus into Judea, as procurator. But the king deprived Joseph of the high priesthood, and bestowed the succession to that dignity on the son of Ananus, who was also himself called Ananus.
Now the report goes that this eldest Ananus proved a most fortunate man; for he had five sons who had all performed the office of a high priest to God, and who had himself enjoyed that dignity a long time formerly, which had never happened to any other of our high priests.
But this younger Ananus, who, as we have told you already, took the high priesthood, was a bold man in his temper, and very insolent; he was also of the sect of the Sadducees, who are very rigid in judging offenders, above all the rest of the Jews, as we have already observed; when, therefore, Ananus was of this disposition, he thought he had now a proper opportunity [to exercise his authority]. Festus was now dead, and Albinus was but upon the road; so he assembled the sanhedrin of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others, [or, some of his companions]; and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned: but as for those who seemed the most equitable of the citizens, and such as were the most uneasy at the breach of the laws, they disliked what was done; they also sent to the king [Agrippa], desiring him to send to Ananus that he should act so no more, for that what he had already done was not to be justified; nay, some of them went also to meet Albinus, as he was upon his journey from Alexandria, and informed him that it was not lawful for Ananus to assemble a sanhedrin without his consent. (24) Whereupon Albinus complied with what they said, and wrote in anger to Ananus, and threatened that he would bring him to punishment for what he had done; on which king Agrippa took the [office of] high priesthood from him, when he had ruled but three months, and made Jesus, son of Damneus, high priest."