Remove the fabric on the chair's arms, being careful not to rip the fabric. Use needle-nose pliers and a screwdriver to pull the staples out of the wood. Open any seam in the fabric and iron it flat. Some bergere chairs are full armchairs, some are more upright, and look like a dining room chair. Full armchairs will have two-part upholstered arms, and the arm fabric will seam to the seat back fabric.
Place the existing fabric over the new upholstery fabric. If the new fabric has a pattern or design you will need to match your pattern so that the pattern stays visually straight on the curved parts of the chair. A good way to understand this is by pinning the new fabric to the chair to align the pattern before cutting the fabric.
Sew the seams, making sure to ease your curves. Easing is a way of making small cuts in the seam allowance so that the fabric can bend around a curve. Replace any old batting on the chair arm with new batting of the same height, or loft. Often you can simply add a layer of fresh batting over the existing stuffing to create the loft, if the existing stuffing is in good condition but just a little flat.
Staple the new fabric over the new batting in the same location where the old upholstery was stapled. Use a staple gun and staple every half inch. Work from the center of the back toward the arm for a traditional bergere armchair. The traditional bergere armchair has an upholstered seat back that wraps around to the arms in an enclosed style. Bergere upright chairs have open wooden arms with upholstery on top of the wood arm. There are many variations on both of these styles that mix how much upholstery is applied over the wood sections of the chair. Match the upholstery pattern of your particular chair.
Fold 1/2-inch of the edge of the fabric under and ease corners from the under side as you work. Trim your fabric as you work so that there isn't a lot of bulk beyond the staple area. Try to position any tucking or gathering along the underside where it won't be seen. Repeat for the seat back front and arms. Repeat for the arm tops.
Hot glue matching trim over the staples. Start at the bottom or least visible location for the seam. Hammer small upholstery tacks every 6-inches hidden as much as possible to further secure the trims.