Cut away the rotted wood from the undamaged portion using a handsaw. Even out the sides and bottom of the rotted area as much as possible with a chisel, removing and disposing of the rotted wood.
Brush the area with isopropyl alcohol to remove any remaining moisture from the wood. Allow the alcohol to evaporate. As it does so, it takes the remaining wood moisture away as well, leaving the wood dry.
Seal off the damaged area by brushing the surface of the wood around the empty space with a solvent of thinned epoxy. Use a foam brush to apply the epoxy to the wood, saturating the wood surrounding the hole until the wood no longer absorbs the epoxy solution. Allow the wood to dry completely.
Brush a thin layer of epoxy putty along the sides and bottom of the hole with the foam brush. Allow the putty to dry to a tacky state and inject the epoxy into the space, filling the space up to a point that's about 1/4 inch from the surface of the wood. Use an injection epoxy in a tube placed in a caulking gun. Run the hose from the nozzle of the tube into the rot area and fill the area with the epoxy, moving the tube to direct the epoxy where needed. Make sure the epoxy packs the area completely, seeping into every crevice of the wood. Wait for the epoxy to dry overnight.
Fill the hole with exterior wood putty by packing the putty into the hole with a putty knife. Fill the hole so that the patch is slightly higher than the surrounding wood surface and overlaps the wood surrounding it slightly, about 1/4 inch. Allow the putty to dry and harden.
Sand the wood putty until it's level with the surrounding wood surface using a medium-grit sandpaper. Sand along the edge of the patch where it overlaps the wood so that it blends in with the surrounding wood surface.
Seal the patched area by brushing it with a light coat of the solvent-thinned epoxy. Coat the surface of the wood surrounding the patch as well to avoid leaving seams along the edge of the epoxy for moisture to seep under. Allow the sealant to dry.
Paint the patch and surrounding wood surfaces to match to conceal the patched area. Prime the area with a suitable primer for the type of paint used, then cover with a top coat. Wait for the paint to dry.