Cut whatever remains of the old seat off the chair frame. Check for tacks holding it on underneath, but don't be concerned about smoothness of the rails.
Lay the end of the rush across one of the rails, snug against the corner and with about 4 inches extending into the space the seat will cover. Wrap the longer end around that rail, bring it up into the center, and draw it across the top of the tail and the other rail that makes that corner. Go around the second rail, close to the corner post of the chair, and under again into the center.
Draw the rush across parallel to the first rail and lift it over the rail at the next corner. Holding the rush close to that corner post, wrap it around the third rail back into the center and then back over the top of the first run and the first rail to come up again in the center from underneath.
Wrap the rush the same way around each of the other corners, running parallel to the last rail, over the top of the next, over the rush and the last rail, and up in the center again after turning the corner.
Continue in this manner around the seat, working inward from the four corners and keeping each round closely parallel to the last. Come up each time in the shrinking center square. Attach new strands, as needed, on the underside and fasten the end there securely.