You'll need to keyframe its orientation following the RB sim. Which is going to require baking the simulation to keyframes, keyframing the right location, deleting other keyframes, and then maybe re-caching a new simulation. That's kind of a pain.
There's something you can do instead, which seems a little less painful to me, involving the use of object constraints: run the simulation to get it cached. Go to the frame before you want it to stop, keyframe animated off. Go to the next frame, make an empty, give it a copy transforms constraint targeting your physics object, then apply that constraint (x button next to the constraint influence, or apply visual transform operation.) That empty is now holding onto the appropriate transform for your object.
Delete the constraint to prevent any dependency loops. On your physics object, enable "animated" and keyframe that property. Your physics object jumps back-- that's okay. Now give the physics object a copy transforms constraint targeting your empty, apply that constraint, keyframe loc/rot, and delete the constraint. You can delete the empty now if you want. You don't have to though, it won't get rendered or affect anything.
Your animated physics object now has the transform it would have gotten from the physics, but as an animated set of loc/rot keyframes.
If you ever change the physics sim, you'll probably have to redo this to get the new orientation.