Season 12 will end on the following dates:
North America: Sunday, February 11 @ 5:00 p.m. PST
Europe: Sunday, February 11 @ 5:00 p.m. CET
Asia: Sunday, February 11 @ 5:00 p.m. KSTSeason 13 will begin about two weeks afterward on the dates below:
North America: Friday, February 23 @ 5:00 p.m. PST
Europe: Friday, February 23 @ 5:00 p.m. CET
Asia: Friday, February 23 @ 5:00 p.m. KST
Source: https://us.battle.net/d3/en/blog/21492188
Check out the Season 13 overview here.