Erect a scaffold or ladder to get to the fascia. Remove the gutter and downspout, if your house has one, and also any overhanging roof shingles.
Pry off damaged fascia board.
Set a circular saw to a 45-degree angle. Remove nails from fascia board.
Cut off the damaged end of the old fascia. Notice where the nail holes are, and cut at a spot where the end falls on a rafter.
Or, put a nailing block at the end of one rafter, if the cut falls past or short of a rafter.
Re-attach the old piece of fascia. Cut a new board to length at a 45-degree angle.
Include the angle of the one end in the measurement so the piece is the correct length.
Join the two mitered ends of the old and new fascia boards, and drill pilot holes at the point where the cuts overlap. Nail into the pilot holes.
Then nail the rest of the new fascia board into the roof rafters.
Paint or stain to match the old fascia. Replace the shingles, gutter and downspout, if necessary.