Remove the seams that hold the sling back to the bars at the top of the chair back and front of the chair frame.
Lay the sling flat and measure its length and width. Flatten the casings where you removed the seams, but do not remove the side hems or the folds at the ends of the casings.
Cut a piece of canvas or other outdoor fabric 1 inch wider and longer than the old sling.
Install a denim sewing machine needle in the sewing machine and thread the machine with nylon upholstery thread. Set the machine to sew a standard zigzag stitch, and sew along the side edges of the sling.
Make 1/2-inch folds on both side edges of the sling and pin them in place. These will be the hems on the sides of the sling. Set the machine to sew a straight stitch with a standard stitch length, and sew the hems in place by stitching 3/8 inch away from the folds.
Set the machine to sew a zigzag stitch, and stitch along the top and bottom edges of the sling. Iron a 1/2-inch fold on both the top and bottom sides.
Thread an upholstery needle with 16 inches of upholstery thread, and knot the ends together. Wrap the top folded edge around the bar at the top of the back of the chair. Hand sew the sling to the chair, stitching 1/8 inch from the folded edge. When you reach the end of the seam, sew back the other way to the starting point, bringing the needle up through the holes you went down through and down through the holes you went up through. The stitches of the second seam will fill in the spaces between the stitches of the first seam. This makes the seam strong enough to support a person's weight. Knot the thread close to the end of the seam, and cut off the excess.
Attach the bottom end of the sling in the same way you attached the top.