He was hit by an ankle injury last season, which restricted him to just five appearances in the latter stages of the campaign.
Despite featuring in all three of Saints' opening Super League fixtures this term, Eastmond's performances have been mixed.
After going unnoticed in the opening day 16-16 draw against Wigan at the Millennium Stadium, he returned to top form with two tries in Saints' 56-22 drubbing of Salford.
He failed to get on the scoresheet
in the 25-18 defeat by Warrington on Friday
at the Halton Stadium and was forced to make an apology to the fans after he walked straight down the tunnel at the final whistle and failed to acknowledge the supporters.
Eastmond follows the likes of Jason Robinson, Shontayne Hape and Chev Walker in swapping the 13-man code for Bath, who currently sit fourth in the Premiership table.
Walker has since returned to rugby league with Hull KR and is currently playing for Bradford Bulls.
"Our record is good with conversions," said Bath skills coach Brad Davis, who also made the move from league to union.
"We can provide him with a platform from which he can not only have a magnificent Bath career but push on to international honours as well."
"But anyone who gets an opportunity to have a look at him, you'll see how dynamic he is as a player."
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