Yes. Take a piece of ply, osb, or particle board that is about 6" larger (min) in dimensions than your cutting board and build a frame around your cutting board. The distance the frame needs to be from your cutting will be the distance from the center of your router minus the distance in from the edge of the cutting board to the center of your groove.
The area in between your cutting board and the frame will be filled with spacers which should be close to the same thickness as your cutting board to keep the router from tipping. There can't be any gaps between the board, spacers, and frame or you could get a wobbly line. The easiest way to build it would be to fix 2 sides to the backer board, install your cutting board and spacers, and the screw the remaining 2 sides down. Work from left to right around the frame and don't use too thick a material for the frame or it could interfere with your router's handles.
This is a really simple jig that only takes about 1/2 hr to make and it can be made from scraps so it costs almost nothing to make.
Someone I consider a master woodworker once told me that a master woodworker is not someone who never makes mistakes. He is someone who is able to cover them up so that no one can tell.