I copied this from e-how...worked fabulously. I had to use the fixed-bar method. I did make a couple of changes, which I included in parenthesis.
1. Examine the chair construction. Note whether the head and seat bars detach to enable you to slide the old sling off the frame.
2. Remove the fasteners that attach the header and seat bars to the chair frame, using a Phillips or flat-head screwdriver, Allen wrench or star driver, depending on the type of fastener holding the chair frame together. Use a power drill to remove dowel pegs if needed.
3. Remove the head and seat bars from the casings. Lay the sling fabric flat. Measure its length from casing seam to casing seam and add 8 inches. Measure its width and add 4 inches.
4. Cut a new piece of fabric to the dimensions from Step 3. Cut a piece of fabric 48 inches long and 34 inches wide, for example, if the original sling measures 40 inches long from casing seam to casing seam and 30 inches wide from left to right.
5. Lay the fabric right-side down on a flat work surface and spread a damp cotton bath towel over it. (I used a steam iron, no towel).The right side of the fabric is prettier, shinier or smoother than the wrong side.
6. Turn the iron to the synthetic setting if you are using recycled PET fabric. Use the high or cotton setting for cotton or hemp fabric.
7. Press the newly-cut piece of sling fabric using a smooth motion, beginning in the center of the fabric and working out to the edges.
8. Remove the towel and fold the sling fabric down 1 inch along each short end. Spread the towel over the sling fabric again and press along the folds.
9. Fold the fabric down 3.5 inches at each end to create the casings for the head and seat bars. Lay the damp towel over the fabric again. Spray the towel with tap water if needed.
10. Press along the folds in a smooth motion, beginning at the center and working to the left side of the fabric, which creates the first casing. Repeat to press from the center to the right side, which creates the second casing.
11. Face one of the long sides of the fabric, with the casings at your right and left. Raise the needle on your sewing machine to its highest position and lift the presser foot.
12. Make sure the fabric is still laying right-side down. The right side is the prettier, smoother or shinier side of the fabric. Move the new sling until the casing on the left side is under the presser foot.
13. Stitch along the right side of the left-hand casing -- which is the side with the most layers -- leaving a 1/4-inch seam allowance between the position of the needle and the open edge of the casing.
14. Repeat to stitch along the left side of the right-hand casing, leaving the same 1/4-inch seam allowance. Turn the sling fabric right-side up.
15. Slide the head and seat bars through the casings. Have a helper hold the seat bar in place, if needed, while you attach the head bar using the original fasteners. Attach the seat bar.
16. Slide one point of the scissors between the head bar and the casing. Cut the casing from the head bar.
17. Slide one point of the scissors between the seat bar and the casing. Cut the casing from the seat bar.
18. Lay the old sling on a flat work surface. Face one of the long sides of the sling, with the casing folds at your right and left. Measure the sling from casing fold to casing fold and add 4 inches. Measure across the short side and add 4 inches as well.
19. Cut a piece of recycled PET, cotton or hemp fabric to the dimensions in Step 3. Lay the fabric right-side down on a flat work surface so that you are facing one of the long sides.
20. Fold the fabric over 1 inch along each short end. Press the fabric using a damp towel and a steam iron on the low or synthetic setting for PET fabric. Use the high or cotton setting for cotton or hemp fabric.
21. Fold the fabric over 1 inch at each end again and press the fabric as stated in the previous step. (I also sewed the long and short ends after pressing them. Thought it would hold the shape longer/better). Mark grommet positions every two inches, beginning 1/2 inch from the left corner closest to you and working toward the left corner farthest from you.
22. Use a hole punch or the point of a pair of scissors to make a hole for each grommet. Sort the grommet tops, which have a longer tube sticking upward, from the bottoms, which have a shorter tube or none at all.
23. Push a top-half grommet through the first hole. Position the top-half grommet cloth-side up in the groove on the grommet anvil.
24. Place a bottom-half grommet curved-side down or tube-side down over the tube of the top-half grommet. Place the grommet-setting tool in the hole of the bottom-half grommet. Strike the grommet-setting tool firmly several times to set the grommet.
25. Repeat until all grommets are set. Position the fabric so you are facing one of the long sides, and the right side of the fabric still faces the table.
26. Measure 6 inches from the center of each grommet at the left side of the sling and make a mark. Repeat for each grommet at the right side of the sling.
27. Set grommets at each marked position, as described in previous steps.
28. Turn the sling so that the right side of the fabric faces the ceiling. Cut two lengths of PET or cotton rope twice as long as the width of the sling, plus 6 to 8 inches.
29. Tie a large knot of your choice in one end of each rope. Have a helper drape the left end of the sling over the head bar of the chair frame, and line up holes in the two rows of grommets.
30. Working from the back of the chair frame, push the rope through the first pair of grommets. Pull until the knot is tight against the back of the sling.
31. Weave the rope back through the second pair of grommets, up through the third and all remaining pairs. Knot the rope after pulling it as tight as you can.
32. Repeat Steps 14 through 16 to attach the sling to the seat bar
Hope this is helpful!