The 23-year-old had been with Province since 2008 where he was initially a winger, but he converted to back-row after impressing in that position.
Bath coach Gary Gold, said: "Nick is a very exciting young player and we are delighted that he has agreed to move to Bath. He is a hard-working, incredibly motivated guy, and we are sure he will fit in with the club ethos we are developing," he told the club's official website.
"We have a very talented and competitive back row at present, and Nick knows he will have to fight for his place, but he is up for the challenge and we can't wait to see the results."
Koster can also play as a forward.