There is no answer to this that is accepted by all scholars. The most usual answer would be that the Final Solution was authorized at the Wannsee Conference of January 1942. However, there are serious issues with this answer.
The Wannsee Conference seems to have been of some major importance in the planning of the Holocaust. There were important leaders present and the notes taken of the meeting suggest that fundamental questions were discussed about what would be done with the Jews.
However, there are clear issues with saying that the Final Solution was first authorized at this conference. For example, there was at least one death camp already operating and another under construction when this conference occurred. There had also been mass shootings of Jews in Russia.
Overall, then, we cannot be sure when the Final Solution was formally authorized. The usual answer is that this happened at the Wannsee Conference, but there are clear problems with that answer.