After the World Cup in 2023, Russell will join Bath from Racing 92.
The Scotland player will leave after the World Cup in France next October after five years with the team.
The 30-year-old, who joined Racing from Glasgow Warriors, has scored over 200 points for the French side.
Russell said it was exciting to be able to join the club.
The direction the club is going in is something I would like to be a part of.
I have learned a lot at Racing 92. It's the right time for a new challenge, in a new league, with the aim of moving Bath back to the top end of the table and competing for domestic and European titles.
After six years with the Warriors, Russell moved to France.
He was a member of the Lions team that traveled to New Zealand and South Africa.
Russell was left out of the Scotland squad for the autumn Tests due to his "form and consistency".
He was back for the match against New Zealand.
"Finn is a world-class player who will greatly add to our squad on and off the field," said Bath head of rugby.
To be able to attract players of Finn's quality to Bath Rugby is a credit to how far we have come.
Since Van Graan took over, Bath have signed a number of players, including ex-Wasps forward Alfie Barbeary and ex-Worcester duo Ted Hill and Ollie Lawrence.
Billy Searle left the club after signing as injury cover.
Chris Jones is a rugby union correspondent for the British Broadcasting Corporation.
Bath have tried and failed to land a big-name fly-half many times over the past decade, so this coup will be warmly received by their supporters.
A number of high-profile players are moving the other way to Russell as the salary cap cuts bite and it is a boost for the league.
Russell will be one of the highest-paid players in the history of the league because his salary is outside the cap.
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